The normal shapes of the symplectic and contact forms over algebras of divided powers
S. Skryabin

 
empty
Normal shapes of symplectic and contact forms]The normal shapes of the symplectic and contact forms over algebras of divided powers





The Lie algebras of Cartan type over the fields of positive characteristic are para­met­rized by differential forms with coefficients in the divided power algebras. The normal shapes of the volume forms have been classified by Tyurin 
and Wilson. 
In this paper, a similar problem is solved for the contact and symplectic forms.


[
    S.M. Skryabin
    
=================



Serge SkryabinNormal shapes of symplectic and contact forms


empty






§ PREFACE TO THE TRANSLATION


This text is a translation into English of a manuscript written in Russian and deposited in 1986 in VINITI archive. The manuscript gives a classification of the differential forms that parametrize the finite-dimensional Lie algebras of alternating hamiltonian and contact types over fields of characteristic p>0. Part of the material covered by this manuscript was published in a journal article <cit.>. 
It was shown there how the classification of 
symplectic forms reduces to certain questions in the linear, or rather semilinear, algebra. For the 
symplectic forms of the 2nd type the classification is then completed quickly. An exposition of these results can be found also in the book of H. Strade <cit.>. 

As to the 
symplectic forms of the 1st type one eventually has to classify objects consisting of a pair of finite-dimensional vector spaces, alternating bilinear forms and flags of subspaces on the two spaces, and a collection of semilinear isomorphisms between the factors of the two flags. This problem is solved in <ref> of the manuscript, and the details have not been published elsewhere.  

The manuscript explains also the classification of contact forms.

It has been the initiative of Dimitry Leites to bring back to life this almost 
forgotten manuscript. The result has come as a pleasant surprise to me. I am 
very grateful to Dimitry Leites for translating, 
and to Dimitry Vorobiev for -coding the manuscript. 
Several minor errors in the original text have been corrected by myself.

The translator has made several changes to the terminology used in the 
manuscript and other papers on modular Lie algebras. The term “hamiltonian 
form” has been replaced by “symplectic form” speaking of the non-degenerate closed differential 2-form, 
and the expression “canonical form of something” by “normal shape of something”.

The class of Cartan type Lie algebras in the setting of the characteristic p>0 theory was introduced in a 1969 paper of A.I. Kostrikin and I.R. Shafarevich <cit.>. Such a Lie algebra consists of special derivations of the free divided power algebra (E) on a finite-dimensional vector space E, and is defined with respect to a decreasing flag  of subspaces of E and a certain differential form ω. Actually the paper <cit.> dealt only with the Lie algebras of Cartan type that admit a standard grading. For their generalization one has to consider a larger class of differential forms with coefficients in the completion (E) of (E), but still related in some way to flags of subspaces of E.

The algebra (E) is presented by generators {x^(r)| x∈ E, r=0,1,…} and relations

    x^(0)=1,    
    (x+y)^(r)=∑_i=0^rx^(i)y^(r-i),    
    (λ x)^(r)=λ^rx^(r),
    x^(r)x^(s)=[ r+s;   r ]x^(r+s),

where x,y∈ E, λ∈ K (the ground field), and r,s∈. The subalgebra () associated with the flag  (see <ref>) is generated by

    {x^(r)| x∈ E_i, r<p^i+1, i=0,1,…}.

The ideals (E)^(k) of (E) are defined inductively by the relation

    (E)^(k)=∑_x∈ E, r⩾ k(E) x^(r)+∑_0<j<k(E)^(j) (E)^(k-j)  for  k>0

starting with (E)^(0)=(E). The ideals ()^(k) of () are similarly defined with the use of only those divided powers x^(r) that lie in ().

A derivation D of (E) is called special <cit.> (or distinguished by Leites and co-authors) if

    D(x^(r))=x^(r-1)D(x)  for all  x∈ E, r>0.

A derivation D of () is called special if the equalities above hold whenever x^(r)∈(). In this case D extends in a unique way to a special derivation of (E), and uniquely extends to a continuous derivation of the completed algebra (E) defined below.

Other relevant notions will be defined in the main body of the paper.



§ DIVIDED POWER ALGEBRAS, DIFFERENTIAL FORMS, COHOMOLOGY,
 AUTOMORPHISM GROUPS


Let K be a field of characteristic p>0. Let E be a finite-dimensional vector space over K and

    :E=E_0⊇ E_1⊇…⊇ 0=E_m=E_m+1=…

a flag of subspaces. Consider the following objects defined in <cit.>: the divided power algebra (E), its subalgebra (), their standard filtrations (E)^(k) and ()^(k), the completion (E) of (E) with respect to this filtration and the closure (E)^(k) of the ideal (E)^(k) in (E). The ideal (E)=(E)^(1) (resp. ()=()^(1)) of the algebra (E) (resp. ()) is maximal and f^p=0 for any f∈(E).

Let W() (resp. W(E)) be the Lie algebra of distinguished (resp. continuous distinguished) derivations of the algebra () (resp. (E)). Let (E) and () be the respective complexes of differential forms. The terms of the standard filtration in W(E) and W() are as follows:

    W(E)_k=(E)^(k+1) W(E),     W()_k=()^(k+1) W().


Let x_1,…,x_n be a basis of E compatible with the flag, and let ∂_i∈W(E) be such that ∂_ix_j=δ_ij (the Kronecker symbol)[In other words, ∂_i is the partial derivative whose action is given by the formula ∂_ix_j^(k)=δ_ijx_j^(k-1) for any k.]. For i=1,2,…,n we set m_i=min{k| x_i∉ E_k}. We will consider the totality of k-forms as the set of (E)-multilinear (resp. ()-multilinear) antisymmetric[In the original manuscript the word “skewsymmetric" was used with the understanding that these are mappings which take zero value whenever any pair of arguments are equal. However, “alternating" is the standard term for such mappings, while “skewsymmetric" should have the same meaning as “symmetric" when p=2. For this reason “alternating" replaces “skewsymmetric" throughout the whole paper.] mappings

    W(E)×…×W(E)_k→(E)  (resp. W()×…× W( )_k→()).

We identify ^k() with the subspace of the forms ∈^k(E) such that 

    (δ_1,…,δ_k)∈()  for all  δ_1,…,δ_k∈ W().

The differential d and the multiplication of forms are defined by the relations 

    d(δ_1,…,δ_k+1)
       =∑_i=1^k+1(-1)^i-1δ_i (δ_1,…,δ_i,…,δ_k+1)+
       3cm+∑_i<j(-1)^i+j([δ_i,δ_j],…,δ_i,…,δ_j,…),
    
    
    (∧')(δ_1,…,δ_k+ℓ)=∑'(π) (δ_π(1),…,δ_π(k)) '(δ_π(k+1),…,δ_π(k+ℓ)),

where ∈^k(E), '∈^ℓ(E) and the sum in (<ref>) runs over all the permutations π of the set {1,…,k+ℓ} such that 
π(1)<…<π(k) and π(k+1)<…<π(k+ℓ).

For every δ∈W(E) and ∈^k(E), define δ∈^k(E) and δ∈^k-1(E) by the formulas 

    (δ)(δ_1,…,δ_k)=δ (δ_1,…,δ_k)-∑_k=1^k(δ_1,…,[δ,δ_i],…,δ_k),
    
    (δ)(δ_1,…,δ_k-1)=(δ,δ_1,…,δ_k-1).

The following identities are well-known

    dδ=δ d,
    δ=δ d+d(δ),
    
    d(∧')=d∧'+(-1)^∧ d',
    δ(∧')=δ∧'+∧δ',
    δ(∧')=(δ)∧'+(-1)^∧(δ').


Let Z^k((E)), resp. B^k((E)), resp. H^k((E)) denote the spaces of closed, resp. exact k-forms, resp. k-cohomology of the complex (E) 

The notation Z^k(()), B^k(()), H^k(()) has a similar meaning. 

In <cit.> continuous mappings →(E), f↦ f^(r) for r=0,1,… have been constructed such that

    (x_1^(α_1)… x_n^(α_n))^(r)=∏_i=1^n(rα_i)!/r! (α_i!)^r x_1^(rα_1)… x_n^(rα_n),
    
    f^(0)=1,
    
    (f+g)^(r)=∑_i=0^rf^(i)g^(r-i),
    
    (λ f)^(r)=λ^rf^(r),
    
    f^(r)f^(s)=[ r+s;   r ]f^(r+s),
    
    (fg)^(r)=0  for   r⩾ p,
    
    (f^(r))^(s)=(rs)!/(r!)^ss! f^(rs),
    
    f^(r)=(E)^(rj),  if f∈(E)^(j),

for any f,g∈(E), λ∈ K, and r,s,α_1,…,α_n⩾ 0. Note a particular case of formula (<ref>):

    (f^(p^k))^(p^ℓ)=f^(p^k+ℓ).

From (<ref>) and continuity of raising to the divided powers it follows that

    df^(r)=f^(r-1)df.

If f∈(), then, generally speaking, f^(r)∉(). We set

    C_k()={f∈()| f^(p^k)∈()}.

In particular, C_k()∩ E=E_k.

The space C_k() has a basis consisting of monomials x_1^(α_1)… x_n^(α_n) with α_i<p^m_i which are not of the form x_i^(p^ℓ), where ℓ+k⩾ m_i.


()^2⊆ C_k() for all k.


Indeed, the monomials x_1^(α_1)x_2^(α_2)… x_n^(α_n) with α_i<p^m_i, other than x_i^(p^ℓ), form a basis of ()^2.

Let (E)^× (resp. ()^×) be the group of invertible elements of the algebra (E) (resp. ()). For each f∈(E) we set

    exp f=∑_i=0^∞ f^(i)∈ 1+(E)⊆(E)^×.

We have  d(exp f)=∑ df^(i)=∑ f^(i-1)df=(exp f) df,  i.e., 

    (exp f)^-1d(exp f)=df.



The mapping u↦ u^-1du gives a surjective homomorphism of the multiplicative group (E)^× onto the additive group Z^1((E)), and the group K^× is its kernel.


Proof.
The fact that the mapping in question is a homomorphism follows from the formula

    (uv)^-1d(uv)=u^-1du+v^-1dv.

Since d(u^-1)=-u^-2du, it follows that u^-1du∈ Z^1((E)). Surjectivity follows from (1.22) and the fact that H^1((E))=0. Finally, if u^-1du=0, then du=0, implying u∈ K.


The mapping exp gives an isomorphism of the additive group (E) onto the multiplicative group 1+(E). We have exp(()^i)=1+()^i for any i⩾ 2.


Proof. The restriction φ of the homomorphism u↦ u^-1du to the subgroup 1+(E) is an isomorphism onto Z^1((E)). Due to (<ref>) we have exp(f)=φ^-1(df). Since d maps (E) onto Z^1((E)) isomorphically, it follows that exp maps (E) onto 1+(E) isomorphically.

If f∈()^i is such that f=gh,  where g,h∈(), then taking into account formula (<ref>) we have

    exp(f)=∑_r=0^p-1f^(r)=∑_r=0^p-11/r! f^r∈ 1+()^i,

and, moreover,

    exp(f)≡ 1+f(1+()^i+1).

Since exp is a homomorphism, it follows that exp(f)∈ 1+()^i and congruence (<ref>) holds for any f∈()^i. For N sufficiently large we have ()^N=0. Now, by the downward induction on i we prove with the aid of (<ref>) that 1+()^i⊆exp(()^i).

Denote by G(E) the group of continuous automorphisms σ of the algebra (E) such that 

    (σ f)^(r)=σ f^(r)  for all   f∈(E),

by G() the group of automorphisms σ of the algebra () such that f^(r) and (σ f)^(r) for any f∈() simultaneously either belong or do not belong to (), and if f^(r),(σ f)^(r)∈(), then (σ f)^(r)=σ f^(r). 

Finally, let G'() be the subgroup of automorphisms σ∈ G() such that σ f-f∈()^2 for all f∈().

The group G(E) leaves the spaces (E)^(k) invariant, the group G() leaves the spaces ()^(k) and C_k()  invariant. Define filtrations of subgroups setting for j⩾ 0


    G(E)_j={σ∈G(E)|σ f-f∈(E)^(j+ℓ)  for all   f∈(E)^(ℓ), ℓ⩾ 0},
    
    G()_j={σ∈ G()|σ f-f∈()^(j+ℓ)  for all   f∈()^(ℓ), ℓ⩾ 0},
    
    G'()_j=G()_j∩ G'().

The groups G()_j and G'()_j are normal in G().


The group G(E) consists of all continuous automorphisms σ of the algebra (E) such that σ W(E) σ^-1=W(E). Similarly,

    G()={σ=()|σ W() σ^-1=W()}.



Therefore, we can define an action of G(E) in the complex (E) and an action of G() in the complex () by setting 

    (σ)(δ_1,…,δ_k)=σ(σ^-1δ_1σ,…,σ^-1δ_kσ).

Directly from the definition we derive the identities

    σ d=dσ,
    σ(∧')=σ∧σ'.



The mapping σ↦(σ x_1,σ x_2,…,σ x_n) establishes a bijection of G(E) onto the set of tuples  (y_1,y_2,…,y_n) such that y_i∈(E) and (∂_iy_j)_1⩽ i,j⩽ n is invertible in (E).
The mapping σ↦(σ x_1,σ x_2,…,σ x_n) establishes a bijection of G() onto the set of tuples (y_1,y_2,…,y_n) such that y_i∈ C_m_i-1() and (∂_iy_j)_1⩽ i,j⩽ n is invertible in ().



The group G() is identified with the subgroup of automorphisms in G(E) leaving the algebra () invariant.


The embedding G()⊆G(E) allows us to consider the action of G() on (E).


The mapping σ↦(σ x_1,σ x_2,…,σ x_n) establishes a bijection of G'() with the set of tuples (y_1,y_2,…,y_n) such that y_i-x_i∈()^2. Under this bijection the subgroup G'()_j corresponds to the tuples such that y_i-x_i∈()^2∩()^(j+1).


Proof.
If σ∈ G'()_j, then σ x_i-x_i∈()^2 by definition of G'(), and σ x_i-x_i∈()^(j+1) by definition of G()_j. 

Conversely, let there be given a tuple (y_1,y_2,…,y_n) such that 
y_i-x_i∈()^2∩()^(j+1). Proposition <ref> and Corollary <ref> show that y_i∈ C_m_i-1().

Further, ∂_iy_j≡δ_ij(), whence (∂_iy_j)≡ 1(), i.e., (∂_iy_j) is invertible. Therefore, there exists a unique σ∈ G() which sends x_i to y_i, and we have to prove that σ∈ G'()_j. For any r<p^m_i we have 

    σ(x_i^(r))-x_i^(r)=y_i^(r)-x_i^(r)=∑_s=1^r(y_i-x_i)^(s)x_i^(r-s).

Since y_i-x_i∈()^2, it follows that (y_i-x_i)^(s)∈()^2 for s⩾ 1, and so σ x_i^(r)-x_i^(r)∈()^2. 

On the other hand, y_i-x_i∈()^(j+1) implies 

    (y_i-x_i)^(s)x_i^(r-s)∈()^(s(j+1)+r-s)⊆()^(r+j)  when   s⩾ 1,

whence σ x_i^(r)-x_i^(r)∈()^(j+r). Set 

    A_k={f∈()^(k)|σ f-f∈()^2∩()^(j+k)}.

It has been proved that x_i^(r)∈ A_r. We claim that A_kA_ℓ⊆ A_k+ℓ. Indeed, if f∈ A_k and g∈ A_ℓ, then 

    σ(fg)-fg=(σ f-f)σ g+f(σ g-g)∈()^2∩()^(j+k+ℓ).

Now it is clear that x_1^(α_1)x_2^(α_2)… x_n^(α_n)∈ A_α_1+α_2+… +α_n, i.e., 

    σ(x_1^(α_1)… x_n^(α_n))-x_1^(α_1)… x_n^(α_n)∈()^2∩()^(j+α_1+… +α_n),

which means that σ∈ G'()_j.

The space E is not stable under the G()-action on (). Let us transfer the G()-module structure to E by means of the isomorphism E≅()/()^(2). Thus, if σ*x, where σ∈ G() and x∈ E, denotes the respective action, then 

    σ x≡σ*x()^(2)  for any  σ∈ G(),   x∈ E.

The differential d induces a G()-equivariant isomorphism 

    ()/()^(2)→^1()/()^1(),

whence we get an isomorphism of G()-modules E≅^1()/()^1().  Proposition <ref> shows that G() acts on E as the group of all linear transformations preserving the flag .

Let us study the action of G() in the cohomology groups. Note that the multiplication of differential forms induces an algebra structure in the space H(()).


The algebra H(()) is isomorphic to the exterior algebra of the space H^1(()). The classes of cocycles x_i^(p^m_i-1)dx_i, where i=1,2,…,n, form a basis of H^1(()).


We will define a G()-invariant decreasing filtration in H^1(()). For k⩾ 0, consider the mappings β_k:C_k()→ Z^1(()):

    β_k(f)=f^(p^k+1-1)df=df^(p^k+1).

Let γ_k:C_k()→ H^1(()) be the composite of β_k and the projection Z^1(())→ H^1(()). The mappings β_k are, clearly, G()-equivariant, and therefore so are the mappings γ_k. The mappings γ_k are, moreover, p^k+1-semilinear. Indeed, for any λ∈ K and f,g∈ C_k(), we have 

    β_k(λ f)=λ^p^k+1β_k(f),
    β_k(f+g)-β_k(f)-β_k(g)=d(∑_i=1^p^k+1-1f^(i)g^(p^k+1-i))∈ B^1(()).

Further, γ_k=C_k+1() since β_k(f)∈ B^1(()) if and only if f^(p^k+1)∈().

Let H^1(())_k be the linear span of the image of γ_k. If K is perfect, then H^1(())_k=γ_k. The G()-equivariance of γ_k implies the G()-invariance of H^1(())_k. If f∈ C_k(), then f^(p)∈ C_k-1() and thanks to (<ref>) we have 

    β_k(f)=df^(p^k+1)=β_k-1(f^(p)),

implying H^1(())_k⊆ H^1(())_k-1. 


For a basis of H^1(())_k one can take the cohomology classes of cocycles {x_i^(p^m_i-1) dx_i| m_i>k}. In particular, H^1(())=H^1(())_0.


Proof.
It follows from the explicit description of the spaces C_ℓ() that 

    C_k()=C_k+1()⊕⟨ x_i^(p^m_i-1-k)| m_i>k⟩_K.

Therefore, γ_k is a vector space over K^p^k+1 whose basis consists of the classes of cocycles

    β_k(x_i^(p^m_i-1-k))=d((x_i^(p^m_i-1-k))^(p^k+1))=dx_i^(p^m_i)=x_i^(p^m_i-1)dx_i ,     m_i>k.



There exist G()-equivariant p^k-semilinear embeddings 

    E_k-1/E_k→ H^1(())_k-1/H^1(())_k

which are bijective if K is perfect.


Proof.
Since E_k-1⊆ C_k-1() and E_k⊆ C_k(), we have γ_k-1(E_k-1)⊆ H^1(())_k-1 and γ_k-1(E_k)=0, i.e., γ_k-1 induces a p^k-semilinear map

    E_k-1/E_k→ H^1(())_k-1→ H^1(())_k-1/H^1(())_k.

If x∈ E_k-1, then it follows from (<ref>) that

    σ*x-σ x∈()^(2)∩ C_k-1()⊆ C_k()⊕⟨ x_i^(p^m_i-k)| m_i>k⟩_k,

whence, by (<ref>),

    γ_k-1(σ*x-σ x)∈⟨γ_k-1(x_i^(p^m_i-k))| m_i>k⟩_K⊆ H^1(())_k ;
    γ_k-1(σ*x)≡γ_k-1(σ x)=σγ_k-1(x)H^1(())_k .

This proves G()-equivariance of the mappings (<ref>). Since 
γ_k-1 takes the basis {x_i| m_i=k} of a complement of E_k 
in E_k-1 to the basis of a complement of H^1(())_k in 
H^1(())_k-1 consisting of the classes of cocycles 
{x_i^(p^m_i-1) dx_i| m_i=k}, the mapping (<ref>) 
is injective (and even bijective if the field K is perfect).


If K is perfect, then G() acts in H^1(()) as the group of all linear transformations preserving all subspaces H^1(())_k for k⩾ 0. Moreover, the group G()_1 acts as the group of those transformations which induce the identity transformations in each quotient space H^1(())_k-1/H^1(())_k.


Proof.
We already know that the group G() leaves the spaces H^1(())_k invariant, and since G()_1 acts on E identically, by the isomorphisms of Proposition <ref> the group G()_1 acts identically on H^1(())_k-1/H^1(())_k.

Conversely, let A be an invertible endomorphism of the space H^1(()) with respect to which the subspaces H^1(())_k are invariant. Let z_i be the class of the cocycle x_i^(p^m_i-1)dx_i, i.e., z_i=γ_m_i-1(x_i). Let us write Az_i=∑ a_ijz_j, where a_ij∈ K and a_ij=0 when m_j<m_i. Set 

    y_i=∑_m_j⩾ m_ib_ijx_j^(p^m_j-m_i),

where the b_ij∈ K are found from the condition b_ij^p^m_i=a_ij. Since A is invertible, it follows that (a_ij)_i,j∈ I_k 0, where I_k={i| m_i=k} for each k, whence (b_ij)_i,j∈ I_k 0, and since y_i≡∑_m_j=m_ib_ijx_j()^(2), the determinant (∂_iy_j)_1⩽ i,j⩽ n is also invertible. Moreover, y_i∈ C_m_i-1(). Therefore, there exists σ∈ G() (Proposition <ref>) such that σ x_i=y_i for all i. Properties of the mappings γ_k imply that 

    σ z_i=σγ_m_i-1(x_i)=γ_m_i-1(σ x_i)=γ_m_i-1(∑ b_ijx_j^(p^m_j-m_i))=∑ b_ij^p^m_iz_j=Az_i.

Finally, if A acts identically on the quotients H^1(())_k-1/H^1(())_k, then a_ij=δ_ij whenever m_j=m_i, and the same holds for the b_ij; hence, y_i-x_i∈()^(2), i.e., σ∈ G()_1.


The group G'() acts trivially on H(()).


Proof.
Since G() acts on H(()) by automorphisms, it suffices to prove that G'() fixesH^1(()) elementwise. From the properties of the mappings γ_k it follows that

    σγ_k(f)=γ_k(σ f)=γ_k(f)+γ_k(σ f-f)=γ_k(f)  for  σ∈ G'(),  f∈ C_k()

since

    σ f-f∈()^2⊆ C_k+1()=γ_k.

It remains to observe that

    H^1(())=H^1(())_0=K·γ_0.


Define the following groups:

    U():={u∈(E)^×| u^-1 du∈^1()},
     B^1():={u^-1 du| u∈()^×}⊆ Z^1(()),
     H^1():=Z^1(())/ B^1().



The mapping u↦ u^-1 du induces a G()-equivariant isomorphism of the multiplicative group U()/()^× onto the additive group H^1().


Proof.
The mapping u↦ u^-1du is obviously G()-equivariant, and by Proposition <ref> it maps U() surjectively onto Z^1(())⊆ Z^1((E)). The pre-image of B^1() under this mapping coincides with ()^×.

By Proposition <ref> and formula (<ref>) we have 

    :={df| f∈()^2}={u^-1du| u∈ 1+()^2}⊆ B^1(())∩ B^1().

For f∈() we have

    (1+f)^-1d(1+f)=∑_i=0^p-1(-1)^if^idf=df+f^p-1df+d(∑_i=1^p-2(-1)^i/i+1 f^i+1)   
    ≡ df-f^(p-1)df    ;
    
    
    df≡ f^(p-1)df B^1().


We have

    Z^1(())= B^1()⊕{∑ c_ix_i^(p^m_i-1)dx_i| c_i∈ K}.

If K is perfect, then Z^1(())= B^1()⊕ dE.


Proof.
Since every element of 1+() is uniquely expressed  in the form u∏_k<m_i(1+a_ikx_i^(p^k)), where a_ik∈ K, u∈ 1+()^2, and since the mapping v↦ v^-1dv isomorphically maps 1+() onto B^1(), we see thanks to (<ref>) that

    B^1()=⊕{∑_k<m_i(a_ikdx_i^(p^k)-a_ik^p dx_i^(p^k+1))| a_ik∈ K}.

The proof is completed by comparing with the equality

    Z^1(())=⊕⟨ dx_i^(p^ℓ)|ℓ⩽ m_i⟩ .



Every element of U() is uniquely represented in the form exp(∑ c_ix_i^(p^m_i))· f, where c_i∈ K, f∈()^×; and if K is perfect, then also in the form exp(x)· f, where x∈ E and f∈()^×.


Note that u () is a subcomplex of (E) for any u∈ U() since for ∈() we have

    d(u)=u(d+u^-1du∧)∈ u ().



For u∈ U()() the complex u () is acyclic.


Proof. By Corollary <ref> we may assume that u=exp(∑ c_ix_i^(p^m_i)). 
Multiplication by u defines an isomorphism of spaces ()≃ u (). Let us transfer the differential of the complex u () by means of this isomorphism. The new differential d' on () is  as follows:

    d'=d+∑ c_ix_i^(p^m_i-1)dx_i∧.

Direct calculation of the cohomology of the complex ((),d') has been performed in <cit.>.


The group G'() acts trivially in H^1(), and therefore also in U()/()^×.


Proof. If σ∈ G'(), then σ x_i-x_i∈()^2 and 

    σ x_i^(p^m_i)-x_i^(p^m_i)=∑_s=1^p^m_i(σ x_i-x_i)^(s)x_i^(p^m_i-s)∈()^2.

By (<ref>) σφ-φ∈⊂ B^1() for each cocycle φ=∑ c_i dx_i^(p^m_i) with c_i∈ K. Therefore, by Proposition 1.15, σφ≡φ B^1() for all φ∈ Z^1(()).


The group G'() leaves invariant every complex u (), where u∈ U().


Let us construct a G()-invariant decreasing filtration in H^1(). Denote by

    d:()→ H^1()

the composition of d:()→ Z^1(()) and the projection Z^1(())→ H^1(). Set 

    H^1()_k= dC_k().


If K is perfect, then H^1()_k= dE_k.


Proof.
Since d(()^2)=⊆ B^1(), it follows from Proposition <ref> that

    H^1()_k={ ∑ a_iℓ  dx_i^(p^ℓ)| k+ℓ<m_i, a_iℓ∈ K}.

In view of (<ref>) we have a_iℓ  dx_i^(p^ℓ)=b_iℓ  dx_i, where the elements b_iℓ are determined from the equation b_iℓ^p^ℓ=a_iℓ. Thus,

    H^1()_k={ ∑_m_i>kc_i  dx_i| c_i∈ K}= dE_k.


If K is perfect, then the G()-modules E_k-1/E_k and H^1()_k-1/ H^1()_k are isomorphic.


Proof. By Lemma <ref> and Proposition <ref> d induces isomorphisms 

    E_k-1/E_k≅ H^1()_k-1/ H^1()_k.

Let us check that they are G()-equivariant. If x∈ E_k-1 and σ∈ G(), then

    σ*x-σ x∈()^(2)∩ C_k-1()=()^2⊕⟨ x_i^(p^ℓ)|ℓ⩾ 1, ℓ+k-1<m_i⟩_K.

Since d(()^2)=⊆ B^1() and d(ax_i^(p^ℓ))≡ a^p^-ℓdx_i B^1() by (<ref>), it follows that 

    d(σ*x-σ x)∈ dE_k= H^1()_k;
     d(σ*x)≡ d(σ x)≡σ d(x) H^1()_k.
    115pt


(K is perfect).
Two elements of H^1()_k-1 H^1()_k are conjugate under G()_1 if and only if their images in H^1()_k-1/ H^1()_k coincide.


Proof. By Proposition <ref> G()_1 acts trivially on H^1()_k-1/ H^1()_k. Therefore, if two elements of H^1()_k-1 are conjugate under G()_1, then their images in H^1()_k-1/ H^1()_k must coincide. Further, any element of H^1()_k-1 is represented uniquely in the form dx, where x∈ E_k-1 and dx∉ H^1()_k if and only if x∉ E_k.

Let x,y∈ E_k-1 E_k. Assume that dx- dy∈ H^1()_k. Then x-y∈ E_k. Let us show that dx and dy are conjugate under G()_1. Proposition <ref> implies the existence of σ∈ G()_1 such that σ x=x+(y-x)^(p). Taking into account (<ref>) we get

    σ dx= d σ x= dx+ d (y-x)^(p)= dx+ d(y-x)= dy.


Consider the following Lie subalgebras in W():

    ()   ={δ∈ W()|δ C_k()⊆ C_k()  for all  k⩾ 0},
    '()   ={δ∈ W()|δ()⊆()^2}⊆(),
    ()_j   =()∩ W()_j,
    '()_j   ='()∩ W()_j.



  ()=∑ C_m_i-1() ∂_i ;  '()=()^2 W().


Proof. Let δ=∑ f_i∂_i∈ W(). If δ∈() (resp. δ∈'()), then f_i=δ x_i∈ C_m_i-1() (resp. f_i∈()^2) since x_i∈ C_m_i-1().

Conversely, let f_i∈ C_m_i-1() for all i. In particular, since f_i∈() for all i, it follows that δ(()^(2))⊆()^2⊆ C_ℓ() for any ℓ. Since C_k()⊆⟨ x_j| m_j>k⟩_K+()^(2) and δ x_j=f_j∈ C_k() for any k<m_j, it follows that δ C_k()⊆ C_k(), i.e., δ∈(). If, moreover, f_i∈()^2 for all i, then δ()⊆()^2 and δ∈'().

(cf. <cit.>)
For j⩾ 1, given σ∈ G'()_j we can select δ∈'()_j, and the other way round, given δ∈'()_j we can select σ∈ G'()_j so that 

    σ x=x+δ x  for all   x∈ E  and
    
    (σ-𝕀-δ)(()^(ℓ))⊆()^(j+ℓ+1)  for all  ℓ⩾ 0.



Proof. Let σ∈ G'()_j. Then f_i=σ x_i-x_i∈()^2∩()^(j+1), whence δ=∑ f_i∂_i∈'()_j. By construction σ x=x+δ x for every x∈ E. If r<m_i, then

    σ x_i^(r)-x_i^(r)-δ x_i^(r)=(x_i+f_i)^(r)-x_i^(r)-x_i^(r-1)f_i=∑_s=2^rx_i^(r-s)f_i^(s).

Since 

    x_i^(r-s)f_i^(s)∈()^(r-s)()^((j+1)s)⊆()^(r+js)⊆()^(r+j+1)    when s⩾ 2,

 it follows that

    σ x_i^(r)-x_i^(r)-δ x_i^(r)∈()^(r+j+1).


Denote

    A_k={h∈()^(k)|σ h-h-δ h∈()^(j+k+1)}.

Thus, x_i^(r)∈ A_r. We have A_kA_ℓ⊆ A_k+l since

    σ(gh)-gh-δ(gh)
    =(σ g-g)(σ h-h)+g(σ h-h-δ h)+(σ g-g-δ g)h∈
    ∈()^(2j+k+ℓ)+()^(j+k+ℓ+1)+()^(j+k+ℓ+1)⊆()^(j+k+ℓ+1)

for g∈ A_k and h∈ A_l. Hence,

    (σ-𝕀-δ)(x_1^(α_1)x_2^(α_2)… x_n^(α_n))⊆()^(j+α_1+α_2+…+α_n+1),

implying (<ref>).

Conversely, if δ∈'()_j, then δ x_i∈()^2∩()^(j+1), and by Proposition <ref> there exists a σ∈ G'()_j that sends x_i to x_i+δ x_i. Again, σ x=x+δ x for any x∈ E. Repeating the above arguments we get (<ref>).


Let ψ∈ u ()^(r) ^k() and σ∈ G'()_j, where u∈ U() and j⩾ 1. Then

    σψ-ψ∈ u ()^(j+r) ^k().

Moreover, if δ∈'()_j is connected with σ by relation 
(<ref>), then 

    σψ-ψ-δψ∈ u ()^(j+r+1) ^k().



The proof is standard.



§ THE G'()-ORBITS OF SYMPLECTIC FORMS


For each form ∈ u ^2() define a homomorphism of ()-modules i_:W()→ u ^1() by the rule i_(δ)=δ. A symplectic form corresponding to the flag  is a closed form ∈ u ^2() for any u∈ U() such that i_ is an isomorphism. Note that if

    =u∑_i<jh_ijdx_i∧ dx_j

where h_ij∈(), then i_ is bijective if and only if (h_ij) is invertible in ().

Denote by () the set of all symplectic forms corresponding to the flag . The forms belonging, respectively, to

    ^1()=()∩()  and  ^2()=()^1()

will be called symplectic forms of the first (resp. second) type.

The action of G() in (E) leaves invariant the sets (), ^1(), and ^2(). The group G'() preserves the sets of symplectic forms in each complex u (), where u∈ U() (Corollary <ref>).

In what follows we will write briefly =(), =(), U=U(), =(), and so on. Note that if ,'∈^2 are closed forms such that  is symplectic and '-∈^2, then ' is also symplectic.


Any form ψ∈^1, where u∈ U, can be expressed as df+ψ,  where f∈ and ψ∈^1.


Proof. Let us write ψ=u(dx+φ),  where x∈ E and φ∈^1. We have

    ψ=d(ux)+u(φ-xu^-1du).



Any form ψ∈^(j+1)^1, where u∈ U, j⩾ 1, can be expressed as df+ψ,  where f∈ u(^2∩^(j+2)) and ψ∈ u(^2∩^(j+1)) ^1.


Proof. Let ψ=u∑ f_i dx_i, where f_i∈^(j+1). Then

    ψ=d(u∑ f_ix_i)-u( ∑ x_i df_i+∑ x_if_iu^-1du).

Obviously, x_if_i∈^2∩^(j+2) and, since df_i∈^(j)^1, we have x_i df_i∈(^2∩^(j+1)) ^1.


Let ∈^k, where u∈ U, be a closed form. Let σ∈ G'_j and δ∈'_j, where j⩾ 1, be related by condition (<ref>). Then there exists φ∈^k-1 such that

    σ-=dφ  and  φ-δ∈^(j+2)^k-1.



Proof.
Suppose first that  is an exact form, i.e., =dψ, where 
ψ∈^k-1. Adjusting ψ by a coboundary we may assume 
thanks to Lemma <ref> that ψ∈^k-1. Setting 

    φ=σψ-ψ-d(δψ),

we get σ-=d(σψ-ψ)=dφ. By (<ref>) δ=δ dψ=δψ-d(δψ), whence, in view of Lemma <ref>,

    φ-δ=σψ-ψ-δψ∈^(j+2)^k-1.


If u∉, then all closed forms are exact (by Proposition <ref>), and so the lemma is proved in this case. 

Let u=1. For every integer ℓ⩾ 0, denote by A^ℓ the subspace of all forms ∈ Z^ℓ() for which the lemma holds. Set A=⊕_ℓ=0^nA^ℓ. We have proved already that B()⊆ A. We claim that A is a subalgebra of . Indeed, let ∈ A^k and '∈ A^ℓ, so that

    σ-=dφ,  σ'-'=dφ'  for some  φ∈^k-1,  φ'∈^ℓ-1

such that

    φ-δ∈^(j+2)^k-1  and  φ'-δ'∈^(j+2)^ℓ-1.

Then 

    σ(∧')-∧'=(σ-)∧σ'+∧ (σ'-')=d(φ∧σ'+(-1)^k∧φ'),

and also

    φ∧σ'+(-1)^k∧φ'-δ(∧')=
    
    =(φ-δ)∧σ'+(-1)^k∧(φ'-δ')+(δ)∧(σ'-').

Since δ∈_j'⊆^(j+1)W, we have δ∈^(j+1) ^k-1. By Lemma <ref> σ'-'∈^(j) ^ℓ. Making use of (<ref>) as well, we see that the element given by the expression (<ref>) lies in ^(j+2) ^k+ℓ-1. Thus, ∧'∈ A^k+ℓ.

Let us show that Z^1()⊆ A. For f∈, we have

    σ df^(p)-df^(p)=d((σ f)^(p)-f^(p))=dh,

where h=∑_r=0^p-1f^(r)(σ f-f)^(p-r)∈() since σ f-f∈^2. Moreover,

    1cm
    h-δ df^(p)=h-δ f^(p)=h-f^(p-1)δ f=
    
    =f^(p-1)(σ f-f-δ f)+∑_r=0^p-2f^(r)(σ f-f)^(p-r).

Thanks to (<ref>) we see that 

    f^(p-1)(σ f-f-δ f)∈^(p+j+1)  and   
    f^(r)(σ f-f)^(p-r)∈^(r+(p-r)(j+1)).

So expression (<ref>) is definitely in ^(j+2), i.e., df^(p)∈ A.

But every 1-cocycle is congruent modulo the space of coboundaries to an appropriate cocycle df^(p). This proves the inclusion Z^1(())⊆ A.

It follows from Proposition <ref> that the algebra Z() is generated by B() and Z^1(). It is clear now that A=Z().


If ∈^k is a closed form, and if σ∈ G'_j,  where j⩾ 1, then σ-=dφ for a suitable form φ∈^(j+1) ^k-1.



Let ,'∈^2,  where u∈ U, be two symplectic forms such that '-=dφ,  where φ∈^(j+1) ^1 and j⩾ 1. Then there exists σ∈ G_j' such that '-σ=dφ for some φ∈^(j+2) ^1.


Proof.
Adjusting φ by a coboundary we may assume thanks to Lemma <ref> that 

    φ∈ u(^2∩^(j+1)) ^1.

Since i_ is an isomorphism of -modules, we can find δ∈(^2∩^(j+1))W=_j' such that δ=φ. Proposition <ref> ensures the existence of a σ∈ G_j' such that (<ref>) holds. By Lemma <ref> σ-=dψ for some ψ∈^1 such that 

    ψ-δ=ψ-φ∈^(j+2) ^1.

Besides, '-σ='--(σ-)=d(φ-ψ).

[It should be added to the hypothesis that here j≥1. In fact, G'_0=G'=G'_1.]

The G'_j-orbit of a symplectic form ∈^2,  where u∈ U, consists of the forms +dφ such that φ∈^(j+1) ^1.


Proof.
By Corollary <ref> the orbit G'_j is contained in the indicated set of forms. Conversely, let us prove that for any symplectic form ∈^2 and an integer k⩾ 1 we have 

    {+dφ|φ∈^(k+1) ^1}⊆ G'_k

by the downward induction on k. For k sufficiently large we have ^(k+1)=0 and the set in the left-hand side of inclusion (<ref>) consists of a single form , so that everything is obvious.

Suppose that (<ref>) has been proved for k=j+1,  where j⩾ 1. Let φ∈^(j+1) ^1. By Lemma <ref> applied to  and '=+dφ we can find σ∈ G'_j and φ∈^(j+2) ^1 such that '-σ=dφ. Hence, '∈ G_j+1'σ⊆ G_j' by the induction hypothesis.


For p>2 symplectic forms ,'∈^2,  where u∈ U, are conjugate under G' if and only if their images in ^2/^2 and in H^2() coincide.
 

Proof.
Since the mappings ^2→^2/^2 and Z^2()→ H^2() are G'-equivariant and G' acts trivially in the space ^2/^2 because G'⊆ G_1, and in the space H^2() by Propositions <ref> and <ref>, it follows that the images of conjugate forms in these spaces coincide. 

Conversely, let  and ' be cohomologous and their initial terms coincide. In particular, '-=dψ,  where by Lemma <ref> we may assume that ψ∈^1. Let ψ=u∑ b_ijx_i dx_j+ψ, where b_ij∈ K and ψ∈^(2)^1. Then 

    dψ≡ u∑ b_ijdx_i∧ dx_j^2.

Since the images of  and ' in ^2/^2 coincide, we have b_ij=b_ji. Set

    φ:=ψ-d(u∑_i<jb_ijx_ix_j+u∑ b_iix_i^(2))=ψ-u(∑_i<jb_ijx_ix_j+∑ b_iix_i^(2))u^-1du.

Then '-=dφ, but now φ∈^(2)^1. It remains to use Proposition <ref> with j=1.

(p>2).
Any symplectic form ∈^2 is conjugate under G' to a unique form of the following expression, where a_ij, b_ij∈ K:

    [ ∑_i<j (a_ij+b_ijx_i^(p^m_i-1)x_j^(p^m_j-1)) dx_i∧ dx_j                                         when  u=1,  or;                                   d(u∑_i<ja_ijx_idx_j)                                            when  u∈ U. ]



We will use the results just obtained for a preliminary description of the orbits of symplectic forms with respect to the whole group G.

Let K be perfect of characteristic p>2. Since G' is normal in G, the group G acts on the set ()/G', and there is a bijection

    ()/G≅(()/G')/G.

Denote by  the subset of bivectors in  corresponding to the non-degenerate bilinear forms on the dual space E^*. There is a mapping

    ()⊆^2(E)→^2(E)/ (E)^2(E)≅^2/ ^2≅⋀^2(^1/ ^1)≅

which is G-equivariant and whose image coincides with . Since G' acts trivially in E, we get a mapping

    ()/G'→ .

The projection Z^2()→ H^2() together with the isomorphism H^2()≅⋀^2(H^1()) (Proposition <ref>) give rise to a G-equivariant mapping

    ^1()⊆ Z^2()→⋀^2H^1().

Since G' acts trivially on H^1() (Proposition <ref>), we get a mapping

    ^1()/G'→⋀^2H^1().

Consider the G-equivariant mapping

    ^1()/G'→×⋀^2H^1().

Its injectivity is equivalent to the conclusion of Proposition <ref>, while surjectivity follows from consideration of the forms explicitly written out in Corollary <ref>. Thus, the mapping (<ref>) is a bijection.

Recall (see <ref>) that there is a G-invariant flag of subspaces in H^1(); we will denote it by . Let _ (resp. _) be the subgroup of linear transformations of the space E (resp. H^1()) preserving the flag  (resp. ). The image G of G in GL(E)× GL(H^1()) is contained in the subgroup T of all pairs (A,B)∈_×_ rendering commutative the diagrams

    E_k-1/E_k@>>>H^1()_k-1/H^1()_k
    
    @VAVV@VVBV
    
    E_k-1/E_k@>>>H^1()_k-1/H^1()_k

where horizontal arrows are the isomorphisms of Proposition <ref>, while vertical arrows are induced by A and B, respectively. Let ρ:T→_ be the projection onto the first factor. Then ρ consists of all pairs (𝕀_E,B),  where B∈_ induces the identity transformation in each factor of the flag . By Proposition <ref> ρ⊆G. Also, ρ(G)=_=ρ(T), whence G=T. As a result we get a bijection 

    ^1()/G≅(^1()/G')/G≅(×⋀^2H^1())/T.


If ∈() is such that ∈^2∩ v^2,  where u,v∈ U, then u^-1v∈^×. Therefore, to each symplectic form there corresponds a well-defined class in U/^×. The resulting mapping ()→ U/O^× is G-equivariant. Composing it with the isomorphism of Proposition <ref> we get a mapping ()→ H^1. Under this mapping non-zero elements of the group H^1 correspond precisely to the symplectic forms of the second type. Since G' acts trivially on H^1 (Proposition <ref>), we obtain a map

    ^2()/G'→ H^1{0}.

Consider the G-equivariant map

    ^2()/G'→×( H^1{0}).

Its injectivity follows from Proposition <ref> since H^2()=0 when u∈ U, while surjectivity follows from Corollary <ref>. Thus,  (<ref>) is a bijection. Since G_1 acts trivially in E, we derive from (<ref>) a G-equivariant bijection

    ^2()/G_1→×(( H^1{0})/G_1).

Let us bring the filtration in the space H^1 (see <ref>) into play. By Lemma <ref> the projections H^1_k-1→ H^1_k-1/ H_k^1 induce bijections

    ( H^1_k-1 H^1_k)/G_1→ ( H_k-1^1/ H_k^1){0}.

Proposition <ref> allows us to re-express the target set in the form (E_k-1/E_k){0}. Now the set ( H^1{0})/G_1 is represented as a disjoint union

    ∐_k⩾ 1 ((E_k-1/E_k){0}).

Substituting this into (<ref>) we get a G-equivariant bijection

    ^2()/G_1→∐_k⩾ 1 (×((E_k-1/E_k){0})).

Since the image of G in GL(E) coincides with the group _, it follows that (<ref>) yields a bijection

    ^2()/G→∐_k⩾ 1 (×((E_k-1/E_k){0}))/_.

Remarks.
1) The condition u∈ U in the definition of a symplectic form was proposed in a somewhat more general form in <cit.>.

2) The results of this section are valid also for p=2 provided that it is additionally[This additional condition concerns only the symplectic forms of the 2nd type.] assumed that all m_i>1 whenever u∉().



§ THE G'()-ORBITS OF CONTACT FORMS


For every form ∈^1(), set

    P_={δ∈ W()|δ d=0},     
    Q_={δ∈ W()|(δ)=0}.

A form ∈^1() will be called a contact form corresponding to the flag  if W()=P_⊕ Q_. The set of all contact forms corresponding to  will be denoted by (). Set

    P_^⊥={ψ∈^1()|ψ(P_)=0},     Q_^⊥={ψ∈^1()|ψ(Q_)=0}.


Recall that i_d(δ)=δ d.


Let ∈(). Then:  P_ and Q_ are free -modules;

 isomorphically maps P_ onto , while i_d isomorphically maps Q_ onto P_^⊥;

Q_^⊥=,    ^1()=⊕ P_^⊥,    f∈() for every f∈^×.


Proof.
From the decomposition W=P_⊕ Q_ it follows that P_ and Q_ are projective finitely generated modules over a local ring . Hence, they are free. Since =Q_, the mapping |_P_ is injective, and, since P_ is a free -module, comparison of dimensions yields (P_)=. Thus,  |_P_ is an isomorphism, implying Q_^⊥=. It follows now from the decomposition W=P_⊕ Q_ that

    ^1=_(W,)=P_^⊥⊕ Q_^⊥=P_^⊥⊕ .

Since i_d=P_, the mapping i_d1mu|_Q_ is injective and i_d(Q_)⊆ P_^⊥. Since ^1= W, we have Q_= P_^⊥, whence i_d(Q_)=P_^⊥.

Obviously, Q_f=Q_. Since d(f)=df∧+fd, the restrictions of the forms d(f) and fd to Q_× Q_ coincide. Therefore, i_d(f) induces an isomorphism of Q_ onto _(Q_,), implying W=P_f⊕ Q_. Thus,  f∈().

The semidirect product  acts on (). Note that in coordinates the condition on a form ∑ f_idx_i to be contact means precisely that the determinant

    ([ ∂_if_j-∂_jf_i           f_i;          -f_j             0 ])

is invertible.
If ,'∈^1 are such that ∈() and '-∈^(2)^1, then d'-d∈^2 and, as is easy to see, '∈(). 


Let ,'∈() and '-=df,  where f∈^2∩^(j+1) with j⩾ 1. There exists σ∈ G_j' such that '-σ∈^(j+1)^1.


Proof.
Since |_P_ is an isomorphism, there is a unique 
∂∈ P_ such that (∂)=1. 
Find some σ∈ G_j' such that (<ref>) is satisfied with 
δ=f∂∈_j'. Since δ d=0 and 
(δ)=f, we have

    δ=δ d+d(δ)=df,

whence (by Lemma 1.25 )

    '-σ=df+-σ=δ+-σ∈^(j+1)^1.



Let ,'∈() be such that '-∈^(j+1) ^1 for some j⩾ 1. Then there exist f∈ 1+^(j+1) and σ∈ G'_j such that fσ'-∈^(j+2) ^1.


Proof. Using Lemma <ref> write '-=dt+χ,  where

    t∈^2∩^(j+2)  and  χ∈(^2∩^(j+1)) ^1.

By Lemma <ref> applied to the forms ' and '-dt we have '-dt-τ'∈^(j+2) ^1 for some τ∈ G_j+1'. Hence

    τ'--χ∈^(j+2)^1.


Let χ=h+ψ, where h∈ and ψ∈ P_^⊥ (Proposition <ref>). Since the projections ^1→ P_^⊥ and ^1→ Q_^⊥ are -module homomorphisms, we have

    h∈^2∩^(j+1)  and  ψ∈(^2∩^(j+1))P_^⊥.

Setting g=(1+h)^-1∈ 1+^(j+1) and φ=(1+h)^-1ψ∈(^2∩^(j+1))P_^⊥, we get by (<ref>)

    gτ'--φ∈^(j+2)^1.


Since i_d:Q_→ P_^⊥ is an isomorphism of -modules, there is a unique derivation

    δ∈(^2∩^(j+1)) Q_⊆_j'

such that φ=δ d. Take σ∈ G_j' for which (<ref>) is satisfied and σ x=x+δ x for all x∈ E (Proposition <ref>). Write =dy+,  where y∈ E, ∈^1. Then

    σ--δ=σ--δ∈^(j+2)^1.

Since (δ)=0, we have 

    δ=δ d+d(δ)=φ,

i.e., σ--φ∈^(j+2)^1. Together with 
(<ref>), this yields 

    gτ'-σ∈^(j+2)^1.

Hence

    σ^-1(g) σ^-1τ'-=σ^-1(gτ'-σ)∈^(j+2)^1.

Here σ^-1(g)∈ 1+^(j+1) and σ^-1τ∈ G_j'.


Two forms ,'∈() such that -'∈^(j+1)^1,  where j⩾ 1, are conjugate under (1+^(j+1))⋊ G_j'.


The proof is obtained from Lemma <ref> by the downward induction on j, taking into account that ^(j+1)=0 for j  sufficiently large.


For p>2 two forms ,'∈() are conjugate with respect 
to the group (1+^(2))⋊ G' if and only if the images of  
and ' in ^1/^1 coincide, and the images of 
d and d' coincide in ^2/^2.  


Proof.
For each free -module M denote by π_M the canonical mapping M→ M/ M, and set π=π_. Define an action of the group ^× on M/ M by assigning to each element f∈^× the multiplication by π(f)∈/≅ K. Оbviously, the mappings π_^1 and π_^2∘ d are G-equivariant. They are also (1+^(2))-equivariant since 

    π_^1(f)=π(f) π_^1(),
    [t]
    π_^2 d(f)   =π_^2(df∧+fd)=π_^1(df)∧π_^1()+π(f) π_^2(d)=
       =π(f)(π_^1(f^-1df)∧π_^1()+π_^2(d)),

and if f∈ 1+^(2), then df∈^1, implying π_^2(d(f))=π_^21mu d(). The group (1+^(2))⋊ G' acts trivially on ^1/^1 and on ^2/^2. Therefore, if ,'∈^1 are conjugate under this group, then π_^1()=π_^1(') and π_^21mu d()=π_^21mu d(').

Conversely, let ,'∈() be such that π_^1()=π_^1(') and π_^21mu d()=π_^21mu d('). Then

    '-=∑ b_ijx_i dx_j+ψ,  where   b_ij∈ K,   b_ij=b_ji,  ψ∈^(2)^1.

Set

    f=∑_i<jb_ijx_ix_j+∑ b_iix_i^(2)∈^2⊂^(2).

By Lemma <ref> applied to  and +df there exists σ∈ G' 
such that +df-σ∈^(2)^1. Then

    '-σ=+df-σ+ψ∈^(2)^1.

By Corollary <ref> the forms ' and σ are conjugate under the group (1+^(2))⋊ G'. Hence, so are ' and ,  too.

(p>2).
Any contact form is G'-conjugate to a form f·∑_i(a_i+∑_jb_ijx_j)dx_i,  where a_i,b_ij∈ K, f∈^× and the matrix (b_ij) can be assumed to be either triangular, or 
skewsymmetric.


(p>2).
Any contact Lie algebra is isomorphic to a contact Lie algebra from <cit.>[The contact forms considered in <cit.> are expressed as ∑ a_idx_i+∑ b_ijx_idx_j with antisymmetric matrix (b_ij). Note, however, that these forms do not satisfy our definition of a contact form when p=2 and n>1, while unitriangular matrices (b_ij) give contact forms in any characteristic, including p=2.].


Let us identify E (resp. ) with the space of linear (resp. bilinear 
antisymmetric) forms on E^*. Denote by  the set of pairs 
(x,b)∈ E× such that

    x 0  and   E^*= x⊕ b.

The direct product of groups  acts on  by the formula

    (λ,x)·(y,b)=(λ y,λ b+λ x∧ y).

The spaces ^1/^1 and ^2/^2 are identified, respectively, with E and  by means of G-equivariant isomorphisms. The mapping which associates to ∈^1 the pair (x,b) in which x corresponds to π_^1() and b∈ corresponds to π_^2(d) sends () to . The G-equivariant mapping ()→ thus obtained induces a map

    ()/((1+^(2))⋊ G')→.

Its injectivity is equivalent to the conclusion of Proposition <ref>, while surjectivity follows from consideration of the forms written out in Corollary <ref>. The group ^× acts on 

    ()/((1+^(2))⋊ G').

In view of (<ref>) the corresponding action on  is performed by means of a surjective homomorphism ^×→. Thus,  bijection (<ref>) yields a bijection

    ()/(')→/().

The group G acts on E as _. This leads to a bijection

    ()/()→/(()⋊_).




§ NORMAL SHAPE OF THE ANTISYMMETRIC BILINEAR FORM WITH RESPECT TO A FLAG OF SUBSPACES


Let V be a finite-dimensional vector space over K, and let

    𝒢:0=V_0⊂ V_1⊂…⊂ V_r=V

be a flag[The flag  will be fixed. By a flag we mean a sequence of subspaces which form either increasing or decreasing chain. Equal terms  are allowed in this sequence.]. Denote by _ the subgroup of all linear transformations of V preserving , and by N() the subgroup of transformations in _ inducing the identity transformations in each factor of the flag  (we will use this notation for arbitrary flags).

For a vector space W denote by (W) the totality of flags of subspaces in W of the form 

    W=W_0⊇ W_1⊇…⊇ W_r

where r is the same as for . Define a mapping 

    (V)→∏_i=1^r (V_i/V_i-1)

assigning to a flag V=V_0'⊇ V_1'⊇…⊇ V_r' the collection of flags

    (V_0'∩ V_i+V_i-1)/V_i-1⊇(V_1'∩ V_i+V_i-1)/V_i-1⊇…⊇(V_r'∩ V_i+V_i-1)/V_i-1
 
with i=1,…,r. This mapping is _-equivariant. Since the group N() acts trivially on each set (V_i/V_i-1), we get a mapping 

    (V)/N()→∏_i=1^r (V_i/V_i-1).



The mapping (<ref>) is a _-equivariant bijection. Flags 

    V=V_0'⊇ V_1'⊇…⊇ V_r'  and   V=V_0”⊇ V_1”⊇…⊇ V_r”

are conjugate under _ if and only if 

    (V_i∩ V_j')=(V_i∩ V_j”)  for all  1⩽ i,j⩽ r.



This is an easy fact from linear algebra.

For a bilinear antisymmetric form b∈ denote by M^⊥_b the orthogonal of a subspace M⊆ V with respect to b. For subspaces M,L⊆ V and an arbitrary antisymmetric form the following known identities hold:

    M^⊥⊥=M+V^⊥;  (M+L)^⊥=M^⊥∩ L^⊥;  (M∩ L^⊥)^⊥=M^⊥+L.


Let ^⊥_b denote the flag

    V=V_0^⊥_b⊇ V_1^⊥_b⊇…⊇ V_r^⊥_b.
 
Thus,  ^⊥_b∈(V). The mapping →(V) such that b↦^⊥_b is _-equivariant.

The form b∈ induces non-degenerate pairings 

    V_i∩ V_j-1^⊥_b/(V_i-1∩ V_j-1^⊥_b+V_i∩ V_j^⊥_b)× V_j∩ V_i-1^⊥_b/(V_j-1∩ V_i-1^⊥_b+V_j∩ V_i^⊥_b)→ K.

Indeed,

    (V_i∩ V_j-1^⊥_b)^⊥_b∩ V_j∩ V_i-1^⊥_b=(V_i^⊥_b+V_j-1)∩ V_j∩ V_i-1^⊥_b=(V_i^⊥_b∩ V_j+V_j-1)∩ V_i-1^⊥_b=V_i^⊥_b∩ V_j+V_j-1∩ V_i-1^⊥_b.

With the help of isomorphisms

    (V_i∩ V_j-1^⊥_b+V_i-1)/(V_i∩ V_j^⊥_b+V_i-1)≅ V_i∩ V_j-1^⊥_b/(V_i∩ V_j^⊥_b+V_i-1∩ V_j-1^⊥_b),
    
    (V_j∩ V_i-1^⊥_b+V_j-1)/(V_j∩ V_i^⊥_b+V_j-1)≅ V_j∩ V_i-1^⊥_b/(V_j∩ V_i^⊥_b+V_j-1∩ V_i-1^⊥_b)

we get non-degenerate pairings

    (V_i∩ V_j-1^⊥_b+V_i-1)/(V_i∩ V_j^⊥_b+V_i-1)×(V_j∩ V_i-1^⊥_b+V_j-1)/(V_j∩ V_i^⊥_b+V_j-1)→ K.


Denote by X the set of tuples ((_i)_1⩽ i⩽ r,(b_ij)_1⩽ i,j⩽ r),  where _i∈(V_i/V_i-1), say _i is a flag

    V_i/V_i-1=U_i0⊇ U_i1⊇…⊇ U_ir ,

and

    b_ij:(U_i,j-1/U_ij)×(U_j,i-1/U_ji)→ K

are non-degenerate pairings such that b_ij(x,y)=-b_ji(y,x) and 
b_ii(x,x)=0. The group

    ∏_i=1^r1mu GL(V_i/V_i-1)≅_/N()

acts on X in a natural way.

To every form b∈ we assign the collection of flags 

    (V_i∩ V_0^⊥_b+V_i-1)/V_i-1⊇ (V_i∩ V_1^⊥_b+V_i-1)/V_i-1⊇…⊇ (V_i∩ V_r^⊥_b+V_i-1)/V_i-1

and pairings (<ref>). We have thus constructed a _-equivariant mapping → X. 
Since N() acts on X trivially, we get a mapping 

    ()/N()→ X.


The mapping (<ref>) is a bijection.


Proof. Let (_i,b_ij)∈ X,  where _i is a flag 

    V_i/V_i-1=U_i0⊇ U_i1⊇…⊇ U_ir .

For every pair i,j, where 1⩽ i⩽ r and 1⩽ j⩽ r+1, pick a basis {e _ij^k| 1⩽ k⩽ n_ij} of the space U_i,j-1/U_ij, assuming that U_i,1mu r+1=0. Choose pre-images e _ij^k of e _ij^k in U_i,j-1 and pre-images e_ij^k of e _ij^k in V_i. Thus,  for each 1⩽ i⩽ r the elements {e _ij^k| 1⩽ j⩽ r+1, 1⩽ k⩽ n_ij} constitute a basis of V_i/V_i-1 compatible with the flag _i, and the elements 

    {e_ij^k| 1⩽ i⩽ r, 1⩽ j⩽ r+1, 1⩽ k⩽ n_ij}

constitute a basis of V compatible with .

Define a bilinear antisymmetric form b:V× V→ K, specifying its values on the basis elements by the rule

    b(e_ij^k,e_ℓ m^n)={[ b_ij(e _ij^k,1mue _ℓ m^n)             if  i=m, j=ℓ,;                         0                otherwise. ].

Since the pairings b_ij are non-degenerate, it follows that the space V_s-1^⊥_b for 1⩽ s⩽ r+1 is spanned by {e_ij^k| j⩾ s}. It is clear now that mapping (<ref>) sends b to the collection (_i,b_ij). Thus,  mapping (<ref>) is surjective.

Let b”∈ be another form which is sent by mapping (<ref>) to the same collection (_i,b_ij). Since the diagram

    @>(4.3)>>X
    
    @VVV@VVV
    (V)@>(4.1)>>∏_i=1^r(V_i/V_i-1)

is commutative, we see that mapping (<ref>) sends the flag 
^⊥_b” to (_i)_1⩽ i⩽ r. By 
Lemma <ref> there exists σ∈ N() such that 
σ^⊥_b”=^⊥_b. Then for b'=σ b” we have 
^⊥_b'=^⊥_b. Now, set 
^⊥=^⊥_b=^⊥_b' and 
V_i^⊥=V_i^⊥_b=V_i^⊥_b' for short. Thus,

    (V_i∩ V_j^⊥+V_i-1)/V_i-1=U_ij.

It remains to show that the forms b and b' are conjugate under N(). Let

    I={(i,j,k)| 1⩽ i⩽ r, 1⩽ j⩽ r+1, 1⩽ k⩽ n_ij}.

It suffices to find elements _ij^k∈ V_i∩ V_j-1^⊥ for all (i,j,k)∈ I such that _ij^k-e_ij^k∈ V_i-1 and

    b'(1mu_ij^k,_ℓ m^n)={[ b_ij(e _ij^k,1mue _ℓ m^n)             if  i=m, j=ℓ,;                         0                otherwise. ].

Indeed, such elements _ij^k form a basis of V compatible with the flag . The linear operator τ:V→ V sending e_ij^k to _ij^k belongs to N() and takes b to b'.

Introduce a linear order on I by setting (i,j,k)<(ℓ,m,n) if and only if

    [                        either            min(i,j)<min(ℓ,m),;                            or        i=min(i,j)=min(ℓ,m)<ℓ,;                            or i=min(i,j)=min(ℓ,m)=ℓ,   j<m,;                            or        min(i,j)=min(ℓ,m)<i<ℓ,;                            or            i=ℓ,   j=m,   k<n. ]


Suppose that (s,t,q)∈ I, the elements _ij^k∈ V_i∩ V_j-1^⊥ such that _ij^k≡ e_ij^kV_i-1 have been found for all (i,j,k)<(s,t,q), and condition (<ref>) holds for all (i,j,k),(ℓ,m,n)<(s,t,q). 

We will construct _st^q∈ V_s∩ V_t-1^⊥ such that _st^q≡ e_st^qV_s-1 and (<ref>) holds for any (i,j,k)<(s,t,q) and (ℓ,m,n)=(s,t,q). In this way all elements _ij^k with required properties will be constructed by induction.

Set

    J    ={(i,j,k)∈ I|(i,j,k)<(s,t,q),   i⩾ t,   j<s},
    
    J'   ={(i,j,k)∈ I|(i,j,k)<(s,t,q),   i<s,   j⩾ t}.

We claim that if (i,j,k)∈ J, then (j,i,ℓ)∈ J' for all 1⩽ℓ⩽ n_ji. We have to verify that (j,i,ℓ)<(s,t,q).

If j<t, then min(i,j)<min(s,t), and therefore (j,i,ℓ)<(s,t,q).

Let j⩾ t. Since j<s, we have t<s and min(s,t)=t. Since (i,j,k)<(s,t,q), but min(i,j)⩾ t, we have either i=t, or j=t and i⩽ s. 

Now, (j,i,ℓ)<(s,t,q) in the first case because min(j,i)=min(s,t)⩽ j<s, and in the second case because j=min(j,i)=min(s,t)<s.

Let

    P=∑_(i,j,k)∈ JK _ij^k,     Q=∑_(i,j,k)∈ J'K _ij^k.

We claim that the pairing b'|_P× Q has no left kernel, and therefore the mapping Q→ P^* induced by b' is surjective. Let us write P=⊕ P_ij and Q=⊕ Q_ij,  where P_ij (resp. Q_ij) is spanned by the vectors _ij^k with (i,j,k)∈ J (resp. J') for the given i,j. By relation (<ref>) b'(P_ij,Q_ℓ m)=0 for i m or jℓ. At the same time the left kernel of the pairing b'|_P_ij× Q_ji is zero, as follows from (<ref>), non-degeneracy of the pairings b_ij, and the fact that _ji^ℓ∈ Q_ji for all ℓ such that 1⩽ℓ⩽ n_ji whenever P_ij 0.

Therefore, there exists x∈ Q such that b'(e_st^q,y)=b'(x,y) for all y∈ P. Set _st^q=e_st^q-x. If (i,j,k)∈ J', then _ij^k∈ V_i∩ V_j-1^⊥⊆ V_s-1∩ V_t-1^⊥, whence x∈ V_s-1∩ V_t-1^⊥. Since e_st^q∈ V_s∩ V_t-1^⊥, we have _st^q∈ V_s∩ V_t-1^⊥ and _st^q≡ e_st^qV_s-1.

The definition of x implies that b'(_st^q,_ij^k)=0 if (i,j,k)∈ J. Let (i,j,k)<(s,t,q), but (i,j,k)∉ J, i.e., either i<t or j⩾ s.

If i<t, then b'(_st^q,_ij^k)=0 because _st^q∈ V_t-1^⊥ and _ij^k∈ V_i⊆ V_t-1. If j>s, then b'(_st^q,_ij^k)=0 because _st^q∈ V_s and _ij^k∈ V_j-1^⊥⊆ V_s^⊥.

Let j=s, but i⩾ t. By condition (i,j,k)<(s,t,q) we must have i=t. Note that for all u∈ V_s∩ V_t-1^⊥ and v∈ V_t∩ V_s-1^⊥ it follows from the definition of b_st that[
More correctly, the equality here follows from the fact that b' is N()-conjugate to b”. Mapping (4.3) takes b' to the given collection (_i,b_ij).] b'(u,v)=b_st(u,v),  where u (resp. v) is the class of u (resp. v) in

    (V_s∩ V_t-1^⊥+V_s-1)/(V_s∩ V_t^⊥+V_s-1)≅ U_s,t-1/U_st   (resp. in  U_t,s-1/U_ts).

Therefore, b'(_st^q,_ts^k)=b_st(e _st^q,e _ts^k).

Thus, formula (<ref>) has been verified for (i,j,k)<(s,t,q) and (ℓ,m,n)=(s,t,q). The proof is complete.


Forms b,b'∈ are conjugate under _ if and only if

    (V_i∩ V_j^⊥_b)=(V_i∩ V_j^⊥_b')

for all i,j such that 1⩽ i,j⩽ r.


Proof.
Let us verify injectivity of the mapping ()/_→(V)/_ induced by the mapping

    →(V),   b↦^⊥_b.

The horizontal arrows in the commutative diagram 

    /_ @>>>(/N())/(_ /N())@>>> X  /∏_i=1^r GL (V_i/V_i-1)
    
    @VVV@VVV@VVV
    (V)/_ @>>>((V)/N())/(_ /N())@>>>∏_i=1^r(V_i/V_i-1)  /∏_i=1^r GL (V_i/V_i-1)

are bijections. Therefore, it suffices to prove that the rightmost downward arrow is an embedding. In other words, we have to verify that any two points of the set X lying over the same point of the set ∏_i=1^r(V_i/V_i-1) are conjugate under ∏_i=1^r GL (V_i/V_i-1).

Let _i:V_i/V_i-1=U_i0⊇ U_i1⊇…⊇ U_ir be flags in (V_i/V_i-1) for i=1,…,r. Let b_ij and b'_ij be non-degenerate pairings U_i,j-1/U_ij× U_j,i-1/U_ji→ K satisfying the identities

    b_ij(x,y)=-b_ji(y,x),     b'_ij(x,y)=-b'_ji(y,x).

There exist nonsingular endomorphisms A_ij of the spaces U_i,j-1/U_ij for 1⩽ i,j⩽ r such that b_ij(x,y)=b'_ij(A_ijx,A_jiy). Pick endomorphisms A_i∈__i which induce A_ij in the quotients U_i,j-1/U_ij. Then the element (A_1,…,A_r) of the group ∏_i=1^rGL (V_i/V_i-1) sends the point ((_i),(b_ij))∈ X to the point ((_i),(b'_ij)).

Note that the classification of bilinear forms relative to a flag all whose factors are of dimension equal to 1 is given in <cit.> . 

The numbers 

    n_ij= V_i∩ V_j-1^⊥ /(V_i∩ V_j^⊥+V_i-1∩ V_j-1^⊥)

are more convenient for us than the dimensions of spaces V_i∩ V_j^⊥.


The _-orbits of bilinear antisymmetric forms on V are in a one-to-one correspondence with the tuples of non-negative integers (n_qt)_1⩽ q,t⩽ r in which n_qq are even, n_qt= n_tq, and ∑_t=1^rn_qt⩽ V_q/V_q-1. The tuples with ∑_t=1^rn_qt=(V_q/V_q-1) correspond to the orbits of non-degenerate forms.


Let 1⩽ k⩽ r. The orbits of the _-action on the set 

    ×((V_k/V_k-1)^* {0})

are in a one-to-one correspondence with the collections of integers ℓ, (n_qt)_1⩽ q,t⩽ r satisfying the following conditions:

    [ n_qt⩾ 0;   n_kℓ 0;   n_qt=n_tq;   1⩽ℓ⩽ r;;      n_qq are even;  ∑_tn_qt=(V_q/V_q-1). ]



Proof.
To (b,f)∈×((V_k/V_k-1)^* {0}) 
we assign the collection n_qt of invariants of the form b (see Corollary <ref>) and the number 

    ℓ=1+max{t| f((V_k∩ V_t-1^⊥_b+V_k-1)/V_k-1) 0}.

Then the numbers n_qt and ℓ depend only on the _()-orbit, and (<ref>) holds. 

Suppose (b',f')∈×((V_k/V_k-1)^* {0}) has the same invariants ℓ and n_qt as (b,f). Let us prove that (b,f) and (b',f') are conjugate under _.

By Corollary <ref> the forms b, b' are conjugate under means of _. Therefore, we may assume at once that b'=b. Denote by X_# the subset of tuples (_i,b_ij)∈ X in which the flags 

    _i:U_i0⊇…⊇ U_ir

end with U_ir=0. Since the form b is non-degenerate, the mapping (<ref>) sends it into some tuple

    κ:=(_i,b_ij)∈ X_#.

By Proposition <ref> we have bijections 

    (×((V_k/V_k-1)^* {0}))/N()→ X_#×((V_k/V_k-1)^*{0}),
    (×((V_k/V_k-1)^* {0}))/_→(X_#×((V_k/V_k-1)^*{0}))/∏_q=1^rGL(V_q/V_q-1).

It remains to show that (κ,f) and (κ,f') are conjugate under ∏  GL (V_q/V_q-1). By definition of ℓ the linear forms f and f' induce non-zero forms f and f' on U_k,ℓ -1/U_kℓ. There exists an invertible endomorphism A_kℓ of the space U_k,ℓ -1/U_kℓ sending f to f'. If k=ℓ, we may assume that A_kk preserves the antisymmetric form b_kk since the symplectic group acts transitively on the set of non-zero linear forms. It follows from the definition of ℓ that the mappings 

    ( f∩ U_k,t-1)/( f∩ U_kt)   → U_k,t-1/U_kt ,
    
    ( f'∩ U_k,t-1)/( f'∩ U_kt)   → U_k,t-1/U_kt

are isomorphisms for tℓ. Using them we find, for t<ℓ, bases {e_t^h| h=1,…,n_kt} and {_t^h| h=1,…,n_kt} of some complements of U_kt to U_k,t-1 such that 

    f(e_t^h)=0,   f'(_t^h)=0,  _t^h-e_t^h∈ U_kt .

There exists an endomorphism A_k∈__k which acts as the identity on U_kℓ, induces A_kℓ on the quotient U_k,ℓ -1/U_kℓ, and sends e_t^h to _t^h. Then A_kf=f', and in the quotients U_k,t-1/U_kt, for tℓ, the endomorphism A_k induces the identity transformations. 

Define endomorphisms A_qt of the spaces U_q,t-1/U_qt to be the identity transformations if (q,t)(k,ℓ) and (q,t)(ℓ,k), while A_ℓ k in the case where ℓ k is determined from the identity

    b_kℓ(A_kℓ^-1u,v)=b_kℓ(u,A_ℓ kv)    
    u∈ U_k,ℓ -1/U_kℓ,  v∈ U_ℓ,k-1/U_ℓ k.

For q k let A_q be an arbitrary endomorphism in __q which induces A_qt in the quotients U_q,t-1/U_qt. Now the element ( A_1,…, A_r)∈∏ GL(V_q/V_q-1) sends f to f' and leaves k unchanged.

We will need an analogue of Proposition <ref> for the case of pairings of two spaces. Let V and W be finite-dimensional vector spaces,

    :0=V_0⊆ V_1⊆…⊆ V_r=V  and  :0=W_0⊆ W_1⊆…⊆ W_r=W

flags of subspaces, B(V,W) the space of bilinear mappings V× W→ K. As earlier, b∈ B(V,W) induces non-degenerate pairings 

    [ V_i∩ W_j-1^⊥_b/(V_i-1∩ W_j-1^⊥_b+V_i∩ W_j^⊥_b)× W_j∩ V_i-1^⊥_b/(W_j-1∩ V_i-1^⊥_b+W_j∩ V_i^⊥_b)→ K;;        (V_i∩ W_j-1^⊥_b+V_i-1)/(V_i∩ W_j^⊥_b+V_i-1)×(W_j∩ V_i-1^⊥_b+W_j-1)/(W_j∩ V_i^⊥_b+W_j-1)→ K. ]


Denote by Y the set of tuples ((_i,_i)_1⩽ i⩽ r,
(b_ij)_1⩽ i,j⩽ r),  where _i∈(V_i/V_i-1) 
and_i∈(W_i/W_i-1); say _i and _i are, 
respectively, flags 

    V_i/V_i-1=T_i0⊇ T_i1⊇…⊇ T_ir,     W_i/W_i-1=U_i0⊇ U_i1⊇…⊇ U_ir,

and b_ij:T_i,j-1/T_ij× U_j,i-1/U_ji→ K are non-degenerate pairings.

The group ∏_i=1^r(GL(V_i/V_i-1)× GL(W_i/W_i-1)) acts on Y.

To a form b∈ B(V,W) we assign the collections of flags 

    V_i∩ W_0^⊥_b+V_i-1/V_i-1⊇ V_i∩ W_1^⊥_b+V_i-1/V_i-1⊇…⊇ V_i∩ W_r^⊥_b+V_i-1/V_i-1,
    
    W_i∩ V_0^⊥_b+W_i-1/W_i-1⊇ W_i∩ V_1^⊥_b+W_i-1/W_i-1⊇…⊇ W_i∩ V_r^⊥_b+W_i-1/W_i-1,

and pairings (<ref>). We have thus constructed a _×_-equivariant mapping B(V,W)→ Y. Since N()× N() acts trivially on Y, we get a mapping

    B(V,W)/(N()× N())→ Y.


The mapping (<ref>) is a bijection.


The proof of Proposition <ref> is repeated almost verbatim. Later on we will apply versions of Propositions <ref> and <ref> for decreasing flags ,  as well.

Consider the set  and the action of the group  on it defined in Section 3. The group GL(V), and therefore the semi-direct product ()⋊ GL(V), also act on .


The orbits under the action of ()⋊_ on  are in a one-to-one correspondence with the collections of integers (k,(n_qt)_1⩽ q,t⩽ r) such that

    [         1⩽ k⩽ r;   n_qt⩾ 0,; n_qq are even,   n_qt=n_tq, ]  ∑_t=1^rn_qt=(V_q/V_q-1)-1   for  q=k,
    (V_q/V_q-1)   for  q k.



Proof. Set Q= f. Assign to (f,b)∈ the collection of numbers

    k=min{q| f(V_q) 0},
    
    n_qt= (Q∩ V_q∩(Q∩ V_t-1)^⊥_b)/(Q∩ V_q∩(Q∩ V_t)^⊥_b+Q∩ V_q-1∩(Q∩ V_t-1)^⊥_b).

They are invariant under the action of the group ()⋊_.

Suppose that (f',b')∈ has the same set of invariants k,n_qt as (f,b). Let us prove that (f,b) and (f',b') are conjugate. Set Q'= f'. The definition of k implies that

    Q∩ V_q /Q∩ V_q-1= Q'∩ V_q /Q'∩ V_q-1=
    
    ( V_q/V_q-1)-1   if q=k
     V_q/V_q-1   if q k.

Hence, (Q∩ V_q)=(Q'∩ V_q) for all q. Therefore, there exists A_1∈_ such that A_1Q=Q'. Then μ A_1f=f' for some μ∈ K^×.

Thus, we may assume that f'=f and Q'=Q. By Corollary <ref> there exists C∈ GL(Q) which preserves the subspaces V_q∩ Q and sends b|_Q to b'|_Q. Let us extend C to some A_2∈_ so that A_2 induce the identity transformation in V/Q. Then A_2f=f and (A_2b)|_Q=b'|_Q.

Thus,  we may assume that f'=f and b'|_Q=b|_Q. Since V=V^⊥_b⊕ Q by the definition of , we have f(e)=1 for a suitable e∈ V^⊥_b. Define g∈ V^* by the formula g(v)=b'(e,v). Then b'=b+f∧ g.



§ THE NORMAL SHAPE OF THE 2ND-TYPE SYMPLECTIC FORMS AND THE CONTACT FORMS


(K is perfect of characteristic p>2).
Any 2nd-type symplectic form corresponding to the flag  is conjugate under G() to a form expressed as

    d(exp(x_i_0)(x_i_1dx_i_1'+…+x_i_sdx_i_s')),

where i_0∈{1,…,n} is an index, {i_1i_1'},…,{i_si_s'} a partition of the set {1,…,n} into pairs, s=n/2.

In order that the form (<ref>) be conjugate to the form

    λ d(exp(x_j_0)(x_j_1dx_j_1”+…+x_j_sdx_j_s”)),

where j_0, {j_1j_1”},…,{j_sj_s”} have the same meaning and λ∈ K^×, it is necessary and sufficient that there exist a permutation of indices which sends i_0 to j_0, every pair in the partition {i_1i_1'},…,{i_si_s'} to a pair occurring in the second partition {j_1j_1”},…,{j_sj_s”}, and maps each subset of indices corresponding to indeterminates[By indeterminates we mean the chosen basis elements x_1,…,x_n∈ E.] of the same height into itself.


Proof. Set 

    V=E^*,     V_q={v∈ V|⟨ v,E_q⟩=0},    =(V_q)_0⩽ q⩽ m.

Then _≅_, E_k-1/E_k≅(V_k/V_k-1)^*, and the set in the right-hand side of (<ref>) is nothing else but

    ∐_k⩾ 1 (×((V_k/V_k-1)^* {0}))/_.

By Lemma <ref> this set and, thanks to bijection (<ref>), the set ^2()/G() are parameterized by the collection of invariants (k,ℓ,n_qt), where 1⩽ k⩽ m, and (ℓ,n_qt) satisfies the relations (<ref>) for r=m. (We assume that E_m=0.)

Let us compute these invariants for the form (<ref>). The mapping (<ref>) sends this form to

    (x_i_1∧ x_i_1'+…+x_i_s∧ x_i_s', x_i_0)∈×(E_m_i_0-1/E_m_i_0),

where x_i_0 is the class of x_i_0 in E_m_i_0-1/E_m_i_0. In particular, k=m_i_0. Let v_1,…,v_n be the basis of V dual to the basis x_1,…,x_n. The partition {i_1i_1'},…,{i_si_s'} determines an involutive permutation i↦ i' on the set {1,…,n}. The elements {v_i| m_i⩽ q} form a basis of V_q and the elements {v_i| m_i'⩾ t} form a basis of V_t-1^⊥. Thus, ℓ=m_i_0'. The classes of elements {v_i| m_i=q, m_i'=t} form a basis of

    (V_q∩ V_t-1^⊥)/(V_q∩ V_t^⊥+V_q-1∩ V_t-1^⊥).

This implies that n_qt={i| m_i=q, m_i'=t}. Observe that the same set of invariants corresponds to a form differing from the form (<ref>) by a non-zero scalar factor.    

Every given collection k,ℓ,n_qt with necessary properties corresponds to a form of shape (<ref>) which can be constructed as follows. Split {1,…,n} into a disjoint union of subsets I_qt, where I_qt⊆{i| m_i=q} and I_qt=n_qt. For all pairs (q,t) with q t choose bijections of I_qt onto I_tq so that the bijections I_qt→ I_tq and I_tq→ I_qt be inverse to each other. Also choose involutive fixed point free permutations on the sets I_qq.

Combining these bijections we get a permutation i↦ i' of the set {1,…,n} which determines a partition into pairs: {i_1i_1'},…,{i_si_s'}. Using the fact that n_kℓ 0, pick any i_0∈ I_kℓ. A form of shape (<ref>) has been thus constructed.

Forms (<ref>) and (<ref>) have identical collections of invariants in the case where m_i_0=m_j_0, m_i_0'=m_j_0”, and the sets I_qt={i| m_i=q, m_i'=t} and J_qt={i| m_i=q, m_i”=t} are of the same cardinality for all q,t (here i↦ i” is the involutive permutation of the set {1,…,n} corresponding to the partition into pairs {j_1j_1”},…,{j_sj_s”}).  If these conditions are satisfied, then there exists a permutation of indices 1,…,n which maps I_qt onto J_qt, sends i_0 to j_0, and, moreover, whenever some i goes to j, the paired index i' goes to j”, that is, the pairs {j_1j_1”},…,{j_sj_s”} in a certain order correspond to the pairs {i_1i_1'},…,{i_si_s'}.

(p>2).
Any contact form corresponding to the flag  is conjugate under G() to a form expressed as

    f1mu(dx_i_0+x_i_1 dx_i_1'+…+x_i_sdx_i_s'),

where f∈()^×, i_0∈{1,…,n} is an index, {i_1i_1'},…,{i_si_s'} are a partition of the set {1,…,n}{i_0} into pairs, s=(n-1)/2. In order that the form (<ref>) be conjugate for some g∈()^× to the form

    g1mu(dx_j_0+x_j_1 dx_j_1”+…+x_j_s dx_j_s”),

where j_01mu,{j_1j_1”},…,{j_sj_s”} have the same meaning[I.e., j_0∈{1,…,n}, and the collection {j_1j_1”},…,{j_sj_s”} is a partition of the set {1,…,n}{j_0} into pairs.], it is necessary and sufficient that there exist a permutation of indices which sends i_0 to j_0, every pair in the partition {i_1i_1'},…,{i_si_s'} to a pair occurring in the second partition {j_1j_1”},…,{j_sj_s”}, and maps each subset of indices corresponding to indeterminates of the same height into itself.


Proof.
Let us pass to V=E^* in the same way as in the proof of Theorem <ref>. By doing this the set in the right-hand side of (<ref>) is interpreted as the set described in Lemma <ref>. Since (<ref>) is a bijection, the orbits of the contact forms under the action of the group  are in a one-to-one correspondence with the collections of invariants k,n_qt subject to constraints (<ref>) with r=m.

Let us compute them for the form (<ref>). Let i↦ i' be the involutive permutation of the set {1,…,n}{i_0} defined by the partition into pairs {i_1i_1'},…,{i_si_s'}. The mapping (<ref>) sends the form (<ref>) to 

    (x_i_0,x_i_1∧ x_i_1'+…+x_i_s∧ x_i_s')∈.

The vectors {v_i| i i_0} constitute a basis of the space Q= x_i_0⊆ V. The vectors {v_i| m_i⩽ q} constitute a basis of V_q and the vectors {v_i| i i_0, m_i'⩾ t} constitute a basis of Q∩ (Q∩ V_t-1)^⊥. The classes of vectors {v_i| i i_0, m_i=q, m_i'=t} constitute a basis of

    Q∩ V_q∩(Q∩ V_t-1^⊥)/(Q∩ V_q∩(Q∩ V_t)^⊥+Q∩ V_q-1∩(Q∩ V_t-1)^⊥).

Thanks to formulas (<ref>), (<ref>) we have

    k=m_i_0,   n_qt={i i_0| m_i=q, m_i'=t}.

The remaining part of the proof is similar  to that of Theorem 5.1.





§ PREPARATION TO THE CLASSIFICATION OF 1ST-TYPE SYMPLECTIC FORMS


Let the field K   be perfect. In a vector space V consider flags of subspaces V(k) of the form

    0=V(0)⊆ V(1)⊆…,    ⋃_k=0^∞ V(k)=V;
    
    0=V(0)⊆ V(1)⊆…⊆ V(∞)=⋃_k=0^∞V(k)⊆ V(∞+1)=V;
    
    0=V(∞)⊆ V(∞+1)=V;
    
    V=V(0)⊇ V(1)⊇…,    ⋂_k=0^∞V(k)=0;
    
    V=V(0)⊇ V(1)⊇…⊇ V(∞)=⋂_k=0^∞ V(k)⊇ V(∞+1)=0;
    
    V=V(∞)⊇ V(∞+1)=0.

We denote the sets of all flags on V of the form (<ref>)–(<ref>), respectively, by

    (V),  (V),  (V),  (V),   (V),   (V).

Let  be one of the flags (<ref>)–(<ref>). Its k-th factor defined as V(k+1)/V(k) for increasing flags and V(k)/V(k+1) for decreasing flags, where k∈{0}∪ for the flags (<ref>) and (<ref>), k∈{0,∞}∪ for the flags (<ref>) and (<ref>), and k=∞ for the flags (<ref>) and (<ref>), will be denoted by Φ_k.

Define a category =_0. Its objects are tuples

    A=(V,,b,W,,c,(ν^k)_k=0,1,2,…),

where V, W are finite-dimensional vector spaces, ∈(V), ∈(W), b:V× V→ K and c:W× W→ K are bilinear antisymmetric forms, where b is non-degenerate, and ν^k:Φ_k→Φ_k are some p^k+1-semilinear isomorphisms. We will define morphisms in  a bit later.

Our goal is to determine objects of the category  up to an equivalence. For this we have to consider more cumbrous objects.

Denote by I_n' (resp. I_n”) the set of sequences =(i_1i_2,…,i_n),  where i_s∈{0,∞}∪ and, if among the indices we encounter ∞ with i_t being the first such index, then t is even (resp. odd), and all the subsequent indices are also equal to ∞.

The set I_0'=I_0” is supposed to consist of the single empty sequence of indices, while I_∞' and I_∞” are the sets of infinite sequences with the property just described. For any i∈ I_n'∪ I_n”, we always assume that i_1 is the first index of the sequence, i_2 is the second one, etc. If i∈ I_n'∪ I_n” and j∈{0,∞}∪, then ij (resp. ji) denotes the sequence

    (i_1,…,i_nj)  (resp. (ji_1,…,i_n)).


The objects of the category _n for n>0 are, by definition, collections

    A=((V, , b)_∈ I_n', (W, )_∈ I_n”, k_1∞, (c)_∈ I_n”, k_1,ℓ_1∞, (ν)_∈ I_n+1', ∈ I_n')

where V and W are finite-dimensional vector spaces among which there are finitely many non-zero ones,  (resp. ) is a flag of subspaces V(q) (resp. W(q)) in V (resp. W) such that

    ∈(V)   for n even, k_n∞;
    (V)   for n even, k_n=∞;
    (V)   for n odd, k_n∞;
    (V)   for n odd, k_n=∞;
    
    ∈(W)   for n even, k_n∞;
    (W)   for n even, k_n=∞;
    (W)   for n odd, k_n∞;
    (W)   for n odd, k_n=∞;
    
    
    b:V× V→ K  and   c:W× W→ K

are non-degenerate bilinear pairings such that

    b(v,v')=-b(v',v)  and   c(w,w')=-c(w',w)

for all v∈ V, v'∈ V1mu, w∈ W, w'∈ W1mu,

    b(v,v)=0  and   c(w,w)=0  for all  v∈ V1mu,   w∈ W1mu,

and

    ν:Φ_k_n+1_(ℓ_1…ℓ_n)^(k_1… k_n)→Φ_ℓ_n_(k_2… k_nk_n+1)^(k_1ℓ_1…ℓ_n-1)

are p^k_1+1-semilinear isomorphisms.

Observe slight differences in the definition of objects of the category _0 in conditions (<ref>), (<ref>).

A morphism of object (<ref>) to an object 

    A=( V, ,  b,  W, ,  c, ν )∈_n,

for n⩾ 0, is a collection

    ((φ)_∈ I_n',(ψ)_∈ I_n”, k_1∞)

of isomorphisms of vector spaces φ:V→ V and ψ:W→ W such that

    φ =  and  ψ =,
     b(φ v,φ v')=b(v,v')  and   c(ψ w,ψ w')=c(w,w')

for all v∈ V, v'∈ V, w∈ W, w'∈ W, and the diagrams 

    Φ_k_n+1_(ℓ_1…ℓ_n-1ℓ_n)^(k_1k_2… k_n)@>ν >>Φ_ℓ_n_(k_2… k_nk_n+1)^(k_1ℓ_1…ℓ_n-1)
    
    @Vφ_(ℓ_1…ℓ_n-1ℓ_n)^(k_1k_2… k_n)VV@Vψ_(k_2… k_nk_n+1)^(k_1ℓ_1…ℓ_n-1)VV
    Φ_k_n+1_(ℓ_1…ℓ_n-1ℓ_n)^(k_1k_2… k_n)@>ν >>Φ_ℓ_n_(k_2… k_nk_n+1)^(k_1ℓ_1…ℓ_n-1)

where the vertical arrows are induced by the indicated mappings, are commutative.  

Observe that all morphisms of the categories considered in this section are equivalences. 

The orthogonal sum of objects (<ref>) and

    A=( V, ,  b,  W,,  c, ν )∈_n

is an object (<ref>) in which

    V= V⊕ V,   W= W⊕ W,

flags  and  are defined by formulas V(q)= V(q)⊕ V(q) and  W(q)= W(q)⊕ W(q), respectively, the pairings b and c by the formulas 

    b( v+ v, v'+ v')= b( v, v')+ b( v, v')  and   c( w+ w, w'+ w')= c( w, w')+ c( w, w')

for all v∈ V, v∈ V, v'∈ V, v'∈ V, w∈ W, w∈ W, w'∈ W, w'∈ W,
and the isomorphisms ν are defined from the commutative diagrams

    [                                      Φ_k_n+1_(ℓ_1…ℓ_n-1ℓ_n)^(k_1k_2… k_n)                                                                                                                Φ_ℓ_n_(k_2… k_nk_n+1)^(k_1ℓ_1…ℓ_n-1);                                                        45-10pt[0pt][0pt]≅                                                                      -45≅; Φ_k_n+1_(ℓ_1…ℓ_n-1ℓ_n)^(k_1k_2… k_n)⊕Φ_k_n+1_(ℓ_1…ℓ_n-1ℓ_n)^(k_1k_2… k_n)                                                                           Φ_ℓ_n_(k_2… k_nk_n+1)^(k_1ℓ_1…ℓ_n-1)⊕Φ_ℓ_n_(k_2… k_nk_n+1)^(k_1ℓ_1…ℓ_n-1) ]

An object will be called decomposable if it is equivalent to the orthogonal sum of some objects A and A in which among the spaces V and W, and also among the spaces V and W, there exist non-zero ones. Otherwise the object will be called indecomposable.

Define categories _m, where m=0,1,…, whose objects are collections

    B=((R, )_∈ I_m+1', ∈ I_m', (d)_∈ I_m+1', (S, )_∈ I_m+1”, i_1∞, ∈ I_m”,
    (e)_∈ I_m+1”, i_1,j_1∞, (μ)_∈ I_m+1', ∈ I_m'),

where R and S are finite-dimensional vector spaces among 
which there are finitely many non-zero ones,  (resp. 
) are flags of subspaces R(q) (resp. S(q)) 
in R (resp. S) such that

    ∈(R)    for m even, j_m∞;
    (R)    for m even, j_m=∞;
    (R)    for m odd, j_m∞;
    (R)    for m odd, j_m=∞;
    
    ∈(S)    for m even, j_m∞;
    (S)    for m even, j_m=∞;
    (S)    for m odd, j_m∞;
    (S)    for m odd, j_m=∞;


    d:   Φ_j_m+1_(j_1… j_m)^(i_1… i_mi_m+1)×Φ_i_m+1_(i_1… i_m)^(j_1… j_mj_m+1)→ K  and
    
    e:   Φ_j_m+1_(j_1… j_m)^(i_1… i_mi_m+1)×Φ_i_m+1_(i_1… i_m)^(j_1… j_mj_m+1)→ K

non-degenerate bilinear pairings such that 

    1em
    .[ d(y,y')=-d(y',y), e(z,z')=-e(z',z),;         d(y,y)=0,         e(z,z)=0, ]}

and

    μ:R_(j_1… j_m)^(i_1… i_mi_m+1)→ S_(i_2… i_m+1)^(i_1j_1… j_m)

are p^i_1+1-semilinear isomorphisms. A morphism of the object (<ref>) to another object

    B=( R, ,  d1mu,  S, ,  e, μ )∈_m

is a collection 

    ((f)_∈ I_m+1', ∈ I_m', (g)_∈ I_m+1”, ∈ I_m”, i_1∞)

of isomorphisms of vector spaces f:R→ R and g:S→ S such that 

    f =,     g =,
     d1mu(φ y,φ y')=d(y,y'),     e(ψ z,ψ z')=e(z,z'),

where the mapping φ:Φ_j_m+1_(j_1… j_m)^(i_1… i_m+1)→Φ_j_m+1_(j_1… j_n)^(i_1… i_m+1) for ∈ I_m+1' is induced by f_(j_1… j_m)^(i_1… i_mi_m+1), the mapping ψ:Φ_j_m+1_(j_1… j_m)^(i_1… i_mi_m+1)→Φ_j_m+1_(j_1… j_m)^(i_1… i_mi_m+1) for ∈ I_m+1” with i_1,j_1∞ is induced by g_(j_1… j_m)^(i_1… i_mi_m+1). The last condition in the definition of a morphism is that the diagrams 

    R_(j_1… j_m)^(i_1i_2… i_m+1)@>μ >>S_(i_2… i_m+1)^(i_1j_1… j_m)
    
    @Vf VV@VVg_(i_2… i_m+1)^(i_1j_1… j_m)V
     R_(j_1… j_m)^(i_1i_2… i_m+1)@>μ >> S_(i_2… i_m+1)^(i_1j_1… j_m)

must be commutative.

The notions of orthogonal sums of objects, decomposable and indecomposable objects are defined by analogy with the categories _n.

We will define the grinding functors

    __n^_n:_n_n  and  __m+1^_m:_m _m+1.


First we will make several preliminary remarks. Let V be a finite-dimensional vector space, =(V(k)) and '=(V[ℓ]1mu) flags of its subspaces. In each quotient Φ_k define a flag denoted by '_Φ_k whose ℓ-th space coincides with the image of V[ℓ]∩ V(k+1) in Φ_k if the flag  is increasing and with the image of V[ℓ]∩ V(k) if the flag  is decreasing. 

Observe that if '∈(V), resp. (V),(V),…, then '_Φ_k∈(Φ_k), resp. (Φ_k), (Φ_k),…. 

Suppose there is given another finite-dimensional vector space U, a flag  in it, and a bilinear pairing b:V× U→ K. Consider the flag ^⊥_b whose spaces coincide with the orthogonals of the spaces of  with the respective indices.

We will say that the flag (^⊥_b)_Φ_k is obtained by transferring the flag  by means of the pairing b to the factor Φ_k. The flags (^⊥_b)_Φ_ℓ are similarly defined.

The pairing b induces non-degenerate pairings

    Φ_ℓ((^⊥_b)_Φ_k)×Φ_k((^⊥_b)_Φ_ℓ)→ K

by the method described in <ref>, see (<ref>). If the pairing b is non-degenerate and ∈(U) or (U), etc., then ^⊥_b∈(V),(V), etc., respectively, and the flag (^⊥_b)_Φ_k is of the same type.

Suppose we are given not a pairing, but a p^t-semilinear isomorphism μ:V→ U, where t is an integer. If  is of one of the types (<ref>)–(<ref>), then the flags μ^-1 and (μ^-1)_Φ_k have the same type. The same is true for the flags μ and (μ)_Φ_ℓ. We will say that the flag (μ^-1)_Φ_k (resp. (μ)_Φ_ℓ) is obtained by transferring the flag  (resp. ) by means of the isomorphism μ to the factor Φ_k (resp. Φ_ℓ). The mapping μ induces a p^t-semilinear isomorphism

    Φ_ℓ((μ^-1)_Φ_k)→Φ_k((μ)_Φ_ℓ).

If, for example, the flags =(U(ℓ)) and =(V(k)) are increasing, then this isomorphism is expressed explicitly as the through mapping

    (μ^-1U(ℓ+1)∩ V(k+1)+V(k))/(μ^-1U(ℓ)∩ V(k+1)+V(k))→
    →(U(ℓ+1)∩μ V(k+1)+μ V(k))/(U(ℓ)∩μ V(k+1)+μ V(k))→
    →(U(ℓ+1)∩μ V(k+1)+U(ℓ))/(U(ℓ+1)∩μ V(k)+U(ℓ)),

where the first arrow is induced by μ and the second one is the canonical isomorphism.

Let us construct the functor __0^_0. Given an object (<ref>), we set __0^_0A=B, where B is the object (<ref>) for m=0, where 

    R^i=Φ_i=V(i+1)/V(i),   S^i=Φ_i=W(i+1)/W(i),   i=0,1,… .

The flag ^i is obtained by transferring the flag  by means of the form b to the factor R^i, i.e.,

    R^i(k)=(V(i+1)∩ V(k)^⊥_b+V(i))/V(i).

Since b is non-degenerate, we have ^i∈(R^i). The flag ^i is obtained by transferring the flag  by means of the form c and adding the spaces S^i(∞), S^i(∞+1), i.e., 

    S^i(k)=(W(i+1)∩ W(k)^⊥_c+W(i))/W(i)  for   k=0,1,… ,
    
    S^i(∞)=(W(i+1)∩ W^⊥_c+W(i))/W(i),   S^i(∞+1)=0.

Thus,  ^i∈(S^i). The pairings 

    d_j^2mu i:Φ_j^i×Φ_i^j→ K,  where   i,j∈{0}∪,

are induced by b and the pairings e_j^1mu i:Φ_j^i×Φ_i^j→ K are induced by c according to the construction 
(<ref>). Finally, μ^i=ν^2mu i.

The functors __n^_n for n>0 are similarly defined. For an object (<ref>) we set __n^_nA=B, where B is the object (<ref>) for m=n, where

    R_(j_1… j_n)^(i_1… i_ni_n+1)=Φ_i_n+1_(j_1… j_n)^(i_1… i_n) ,     S_(j_1… j_n)^(i_1… i_ni_n+1)=Φ_i_n+1_(j_1… j_n)^(i_1… i_n) .

The flag _(j_1… j_n)^(i_1… i_ni_n+1) is obtained by transferring the flag _(i_1… i_n)^(j_1… j_n) by means of the pairing b_(j_1… j_n)^(i_1… i_n) to the factor R_(j_1… j_n)^(i_1… i_ni_n+1). The flag _(j_1… j_n)^(i_1… i_ni_n+1) for j_1∞ is obtained by transferring the flag _(i_1… i_n)^(j_1… j_n) by means of the pairing c_(j_1… j_n)^(i_1… i_n) to the factor S_(j_1… j_n)^(i_1… i_ni_n+1), while for j_1=∞ it is determined by its membership in (S) when n is odd and in (S) when n is even (conditions (<ref>) and (<ref>) imply (<ref>) and (<ref>), respectively). The pairings d_(j_1… j_nj_n+1)^(i_1… i_ni_n+1) and e_(j_1… j_nj_n+1)^(i_1… i_ni_n+1) are obtained by applying construction (<ref>) (conditions (<ref>), (<ref>) imply (<ref>)). Finally, μ_(j_1… j_n)^(i_1… i_ni_n+1)=ν_(j_1… j_n)^(i_1… i_ni_n+1).

If (<ref>) is a morphism of the object (<ref>) to the object (<ref>), then φ (resp. ψ) induces isomorphisms of the factors of the flag  onto the factors of the flag  (resp. isomorphisms of the factors of the flag  onto the factors of the flag ), and the collection of these isomorphisms is a morphism __n^_nA→__n^_n A. This defines the functor __n^_n on morphisms. 

Define the functor __m+1^_m setting __m+1^_mB=A for the object (<ref>), where A is defined as follows (see (<ref>) for n=m+1):

    V_(ℓ_1…ℓ_mℓ_m+1)^(k_1… k_mk_m+1)=Φ_ℓ_m+1_(ℓ_1…ℓ_m)^(k_1… k_mk_m+1),     W_(ℓ_1…ℓ_m)^(k_1… k_mk_m+1)=Φ_ℓ_m+1_(ℓ_1…ℓ_m)^(k_1… k_mk_m+1).

The flag _(ℓ_1…ℓ_mℓ_m+1)^(k_1… k_mk_m+1) is obtained by transferring the flag _(k_2… k_m+1)^(k_1ℓ_1…ℓ_m) by means of the isomorphism μ_(ℓ_1…ℓ_m)^(k_1… k_mk_m+1) to the factor V_(ℓ_1…ℓ_mℓ_m+1)^(k_1… k_mk_m+1). The flag _(ℓ_1…ℓ_mℓ_m+1)^(k_1… k_mk_m+1) is obtained by transferring the flag _(k_2… k_m+1)^(k_1ℓ_1…ℓ_m) by means of the isomorphism μ_(k_2… k_m+1)^(k_1ℓ_1…ℓ_m) to the factor W_(ℓ_1…ℓ_mℓ_m+1)^(k_1… k_mk_m+1) (from (<ref>), (<ref>) it follows that (<ref>), (<ref>) hold). The isomorphisms ν_(ℓ_1…ℓ_m+1)^(k_1… k_m+2) are determined by the construction (<ref>). Finally,

    b_(ℓ_1…ℓ_m+1)^(k_1… k_m+1)=d_(ℓ_1…ℓ_m+1)^(k_1… k_m+1),     c_(ℓ_1…ℓ_m+1)^(k_1… k_m+1)=e_(ℓ_1…ℓ_m+1)^(k_1… k_m+1)

(from (<ref>) it follows that (<ref>), (<ref>) hold).


Objects A, A∈_n are equivalent if and only if so are __n^_nA and __n^_n A.


Proof.
Let A be the object (<ref>), let A be the object (<ref>), and let (<ref>) be an equivalence __n^_nA→__n^_n A. For each pair ∈ I_n' choose arbitrarily isomorphisms σ:V→ V so that σ sends  to  and the induced mappings of factors Φ_i→Φ_i coincide with f_^ i. Similarly, for ∈ I_n”, where k_1∞, choose isomorphisms τ:W→ W so that τ transforms  to  and the induced mappings of factors Φ_i→Φ_i coincide with g_^ i.

Let us transfer the structure of the object A by means of isomorphisms σ and τ to the spaces V and W. We obtain an object, say (<ref>), in which 

    V=V,   W=W,  =,  =.

By construction, the collection (σ,τ) defines an equivalence A→ A and the induced equivalence __n^_n A→__n^_n A is given by the same collection (f,g) as the equivalence __n^_nA→__n^_n A. Therefore, the collection of identity mappings gives an equivalence __n^_nA→__n^_n A.

This implies, first, thanks to condition (<ref>) and the definition of equivalence in the category _n that ν=ν for all ∈ I_n+1', ∈ I_n', i.e., A differs from A only by a collection of pairings.

Second, thanks to (<ref>), the flags _^ i and _^ i in Φ_i that enter the structure of the objects __n^_nA and __n^_n A, respectively, coincide, and the same is true for the flags _^ i and _^ i in Φ_i.

Third, by (<ref>) the pairings b and b (resp. c and c) induce the same pairings 

    Φ_j_^ i×Φ_i_^ j→ K  (resp. Φ_j_^ i×Φ_i_^ j→ K).

By the definition of the grinding functors this means that for any ∈ I_n' such that  (resp. ∈ I_n” such that k_1,ℓ_1∞ and ) the mapping (<ref>) sends b,  b1mu∈ B(V,V) (resp. c,  c1mu∈ B(W,W)) to the same point. By Proposition <ref> there exist φ=N() and φ∈ N() (resp. ψ∈ N() and ψ∈ N()) such that 

    b1mu(φ v,φ v')=B(v,v'),     c1mu(ψ w,1muψ w')=c(w,w')
    for all   v∈ V,   v'∈ V,   w∈ W,   w'∈ W.

For = using Proposition <ref> instead of Proposition <ref> we find φ and ψ with similar properties. The collection of mappings (φ,ψ) determines an equivalence A→ A.

Indeed, conditions (<ref>) and (<ref>) in the definition of equivalence in _n are satisfied by construction. Condition (<ref>) holds since φ (resp. ψ) induce the identity transformations of the factors Φ_i (resp. Φ_i), Therefore, the objects A and A are also equivalent. 

Let V, V,W, W be finite-dimensional vector spaces, μ:V→ W and μ: V→ W semilinear isomorphisms,

    =(V(k)),  =( V(k)),  =(W[1muℓ2mu]1mu),  =( W[1muℓ2mu]1mu)

flags of types (<ref>)–(<ref>) such that  and  are of the same type,  and  are also of the same type. Suppose that we are given collections of linear isomorphisms φ_k:Φ_k→Φ_k and ψ_ℓ:Φ_ℓ→Φ_ℓ of all factors that make sense.

In order for isomorphisms f:V→ V and g:W→ W which transform, respectively,  to  and  to   to exist, induce isomorphisms φ_k and ψ_ℓ of the corresponding factors, and make the diagram

    V@>μ >>W
    
    @VfVV@VVgV
     V@>μ>> W

commutative, it is necessary and sufficient that every φ_k transform (μ^-1)_Φ_k to (μ^-1)_Φ_k , every ψ_ℓ transform (μ)_Φ_ℓ to (μ)_Φ_ℓ , and all the diagrams

    Φ_ℓ((μ^-1)_Φ_k)@>>>Φ_k((μ)_Φ_ℓ)
    
    @VVV@VVV
    Φ_ℓ((μ^-1)_Φ_k)@>>>Φ_k((μ)_Φ_ℓ)

in which horizontal arrows are the mappings (<ref>), vertical arrows are induced, respectively, by φ_k and ψ_ℓ be commutative.  


Proof.
Everything boils down to the case where W=V, μ=id_V, W= V, μ=id_ V. If an isomorphism f:V→ V sending  to ,  to  and inducing isomorphisms φ_k and ψ_ℓ exists, then, as is easy to see, φ_k sends _Φ_k to _Φ_k, ψ_ℓ sends _Φ_ℓ to _Φ_ℓ and the diagrams

    [ Φ_ℓ(_Φ_k)         ≅ Φ_k(_Φ_ℓ);      φ_k↓      ↓ψ_ℓ; Φ_ℓ(_Φ_k)         ≅ Φ_k(_Φ_ℓ) ]

commute.

Conversely, let these properties of the collection (φ_k,ψ_ℓ) be satisfied. Choose a basis of V compatible with both the flag  and the flag . Let e be a basis element. Without loss of generality we may assume all the flags to be increasing. Suppose that 

    e∈ V(k+1)∩ V[ℓ+1],   e∉ V(k),   e∉ V[1muℓ2mu].

Let e be the image of e in Φ_k and e the image of e in Φ_ℓ. Then

    u   =φ_k1mue∈( V(k+1)∩ V[ℓ+1]+ V(k))/ V(k),
    u   =ψ_ℓ2mue∈( V(k+1)∩ V[ℓ+1]+ V[1muℓ2mu])/ V[1muℓ2mu],

where thanks to (<ref>) the image of u in Φ_ℓ(_Φ_k) and the image of u in Φ_k(_Φ_ℓ) correspond to each other. Therefore, there exists u∈ V(k+1)∩ V[ℓ+1] whose image in Φ_k coincides with u and image in Φ_ℓ coincides with u. Assign this vector u to the vector e. The linear mapping f:V→ V defined on the basis elements in this way has the desired properties.


Objects B, B∈_m are equivalent if and only if so are __m+1^_mB and __m+1^_m B.


Proof.
Let B be object (<ref>) and B object (<ref>); let (<ref>) be an equivalence 

    __m+1^_mB→__m+1^_m B.

By conditions (<ref>), (<ref>) and the definition of an equivalence in _m+1 it follows that for every ∈ I_m+1' and ∈ I_m' we can apply Lemma <ref> to the spaces R, R, S_(i_2… i_m+1)^(i_1j_1… j_m), S_(i_2… i_m+1)^(i_1j_1… j_m), the isomorphisms μ, μ, the flags , , _(i_2… i_m+1)^(i_1j_1… j_m), _(i_2… i_m+1)^(i_1j_1… j_m), and the collection of isomorphisms of factors

    φ_ k^:Φ_k→Φ_k ,    ψ_(i_2… i_m+1ℓ)^(i_1j_1… j_m):Φ_ℓ_(i_2… i_m+1ℓ)^i_1j_1… j_m)→Φ_ℓ_(i_2… i_m+1ℓ)^(i_1j_1… j_m) .

Therefore, there exist isomorphisms

    f:R→ R,     g_(i_2… i_m+1)^(i_1j_1… j_m):S_(i_2… i_m+1)^(i_1j_1… j_m)→ S_(i_2… i_m+1)^(i_1j_1… j_m)

which transform the flag  to , the flag _(i_2… i_m+1)^(i_1j_1… j_m) to _(i_2… i_m+1)^(i_1j_1… j_m), induce isomorphisms φ_ k^ and ψ_(i_2… i_m+1k)^(i_1j_1… j_m), and make the diagram (<ref>) commutative. The obtained collection (<ref>) gives an equivalence B→ B since condition (<ref>) coincides precisely with condition (<ref>) applied to the initial equivalence __m+1^_mB→__m+1^_m B.

An object of _n for n>0 (similarly, an object of _m) will be called primitive if all the flags entering its structure do not contain proper non-zero subspaces. We denote by _n (resp. by _m) the full subcategory of all primitive objects in _n (resp. in _m).

The grinding functors preserve primitive objects. If (<ref>) is a primitive object, then its support is the set 

    A= A_1∪_2A,  where  _1A={(,l)∈ I_n'× I_n'| V 0},
    _2A={()∈ I_n”× I_n”| k_1∞, W 0}.

Similarly, the support of a primitive object (<ref>) is 

    B=_1B∪_2B,  where  _1B={()∈ I_m+1'× I_m'| R 0},
    _2B={()∈ I”_m+1× I_m”| i_1∞, S 0}.

To study the primitive objects we introduce a new category . Its objects are collections 

    C=((V,b1mu)_∈ I_∞', (W1mu)_∈ I_∞”, i_1∞, (c1mu)_∈ I_∞”, i_1,j_1∞, (ν1mu)_∈ I_∞'),

where V, W are finite-dimensional vector spaces among which there are finitely many non-zero ones,

    b:V× V→ K  and    c:W× W→ K

are non-degenerate bilinear pairings satisfying identities (<ref>) and (<ref>), and

    ν:V_(j_1j_ 2…)^(i_1i_2i_3…)→ W_(i_2i_3…)^(i_1j_1j_2…)

are p^i_1+1-semilinear isomorphisms. The support of object (<ref>) is the set 

    C=_1 C∪_2C,  where  _1C={()∈ I_∞'× I_∞'| V 0},
    _2C={()∈ I”_∞× I”_∞| i_1∞, W 0}.

The morphisms (equivalences), as well as orthogonal sums, decomposable and indecomposable objects are defined in an obvious way.

Now we will define functors _n. For m⩾ n, denote by __m^_n:_n_m and __m^_n:_n_m the composition of functors __r^_r and __s+1^_s ordered as needed. Suppose that (<ref>) is a primitive object in _n. Consider the sequence of objects for m⩾ n:

    A_m=__m^_nA=((V, , b1mu)_∈ I_m', (W, 1mu)_∈ I_m”, k_1∞,
    (c1mu)_∈ I_m”, k_1,ℓ_1∞, (ν1mu)_∈ I_m+1', ∈ I_m')∈_m ,


    B_m=__m^_nA=((V, 1mu)_∈ I_m+1', ∈ I_m', (b1mu)_∈ I_m+1', (W, 1mu)_∈ I_m+1”, k_1∞, ∈ I_m”,
    (c1mu)_∈ I_m+1”, k_1,ℓ_1∞,(ν1mu)_∈ I_m+1', ∈ I_m')∈_m .

Since the object A_m is primitive, it follows that any space of the flag  is either zero, or coincides with V. Therefore, for any ()∈_1A_m, there exists a unique index q such that V_^ q=Φ_q 0, i.e., ( q, )∈_1B_m, and for this index V_^ q≅ V. Therefore, the mapping 

    ((k_1… k_mk_m+1),(ℓ_1…ℓ_m))↦((k_1… k_m),(ℓ_1…ℓ_m))

gives a bijection _1B_m→_1A_m and, similarly, a bijection _2B_m→_2A_m. Moreover, there are isomorphisms

    V_(ℓ_1…ℓ_m)^(k_1… k_mk_m+1)≅ V_(ℓ_1…ℓ_m)^(k_1… k_m)  for all  ((k_1… k_mk_m+1),(ℓ_1…ℓ_m))∈_1B_m,
    
    W_(ℓ_1…ℓ_m)^(k_1… k_mk_m+1)≅ W_(ℓ_1…ℓ_m)^(k_1… k_m)  for all  ((k_1… k_mk_m+1),(ℓ_1…ℓ_m))∈_2B_m.

Similarly, the mapping

    ((k_1… k_mk_m+1),(ℓ_1…ℓ_mℓ_m+1))↦((k_1… k_mk_m+1),(ℓ_1…ℓ_m))
 
gives bijections _1A_m+1→_1B_m and _2A_m+1→_2B_m1mu, and there are canonical isomorphisms

    V_(ℓ_1…ℓ_mℓ_m+1)^(k_1… k_mk_m+1)≅ V_(ℓ_1…ℓ_m)^(k_1… k_mk_m+1)  for all  ((k_1… k_mk_m+1),(ℓ_1…ℓ_mℓ_m+1))∈_1A_m+1,
    
    W_(ℓ_1…ℓ_mℓ_m+1)^(k_1… k_mk_m+1)≅ W_(ℓ_1…ℓ_m)^(k_1… k_mk_m+1)  for all  ((k_1… k_mk_m+1),(ℓ_1…ℓ_mℓ_m+1))∈_2A_m+1.

The obtained identifications of V_(ℓ_1…ℓ_m)^(k_1… k_m) with V_(ℓ_1…ℓ_mℓ_m+1)^(k_1… k_mk_m+1) and of W_(ℓ_1…ℓ_m)^(k_1… k_m) with W_(ℓ_1…ℓ_mℓ_m+1)^(k_1… k_mk_m+1) are compatible with the pairings, while the identifications of V_(ℓ_1…ℓ_m)^(k_1k_2… k_m+1) with V_(ℓ_1…ℓ_mℓ_m+1)^(k_1k_2… k_m+1k_m+2) and of W_(k_2… k_m+1)^(k_1ℓ_1…ℓ_m) with W_(k_2… k_m+1k_m+2)^(k_1ℓ_1…ℓ_mℓ_m+1) are compatible with the semilinear isomorphisms. Set 

    Z_1   ={()∈ I_∞'× I'_∞|((i_1… i_m),(j_1… j_m))∈_1A_m  for all  m⩾ n},
    
    Z_2   ={()∈ I”_∞× I”_∞|((i_1… i_m),(j_1… j_m))∈_2A_m  for all  m⩾ n}.

To object (<ref>) we assign object (<ref>) in which 

    V=V_(j_1… j_n)^(i_1… i_n) ,     W=W_(j_1… j_n)^(i_1… i_n)

if ()∈ Z_1 (resp. ()∈ Z_2), and we set V=0 (resp. W=0) otherwise. If ()∈ Z_1, then

    ((j_1… j_m),(i_1… i_m))∈_1A_m  for all  m⩾ n
 
by the non-degeneracy of pairings b_(j_1… j_m)^(i_1… i_m), whence (,)∈ Z_1. Similarly, ()∈ Z_2 and j_1∞ imply (,)∈ Z_2. For each ()∈ Z_1 (resp. ()∈ Z_2 with j_1∞) we set 

    b=b_(j_1… j_n)^(i_1… i_n) ,     c=c_(j_1… j_n)^(i_1… i_n) .

If ()∈ Z_1, then 
((i_1i_2… i_m+1),(j_1… j_m))∈_1B_m1mu, 
for all m⩾ n, whence

    ((i_1j_1… j_m),(i_2… i_m+1))∈_2B_m

by bijectivity of the mappings ν_(j_1… j_m)^(i_1i_2… i_m+1). This implies ((i_1j_1… j_m-1),(i_2i_3… i_m+1))∈_2A_m1mu, i.e., ((i_1j_1j_2…),(i_2i_3…))∈ Z_2. Set

    ν=ν_(j_1… j_n)^(i_1i_2… i_n+1).

Thus,  we have bijections _1C→_1A and _2C→_2A, as well as equalities (<ref>) for all ()∈_1C (resp. ()∈_2C). This entails immediately


Let (<ref>) and (<ref>) be two primitive objects in _n to which objects C, C of  correspond. Equivalences A→ A are in a bijective correspondence with equivalences C→ C.


We also have commutative diagrams

    2pt
    [            _n;     __n+1^_n↓ to40pt-30⟶30⟶              ;          _n+1               ]


Let A be an object of . If __m^A is not primitive, then one of the spaces entering the structure of the object __m^A yields under grinding a factor of dimension strictly less than the dimension of the space itself. Therefore, for n sufficiently large the object __n^A will be primitive.

Denote by A the image of __n^A in . Since the diagram (<ref>) is commutative, this image does not depend on n. Thus,  we have defined a functor

    :.



Two objects A, A∈ are equivalent if and only if so are A and A.


This follows from Lemmas <ref>, <ref>, <ref>.


Any object of the category  is equivalent to A for a suitable A∈.


Proof.
Consider an object (<ref>). For n⩾ 0, set

    V_(ℓ_1…ℓ_n)^(k_1… k_m)=⊕_∈ I'_∞ | i_1=k_1,…, i_m=k_m,j_1=ℓ_1,…, j_n=ℓ_nV ,     W_(ℓ_1…ℓ_n)^(k_1… k_m)=⊕_∈ I”_∞ | i_1=k_1,…, i_m=k_m,j_1=ℓ_1,…, j_n=ℓ_nW ,

where m=n or n+1. Define the pairings

    b_(ℓ_1…ℓ_n)^(k_1… k_n):V_(ℓ_1…ℓ_n)^(k_1… k_n)× V_(k_1… k_n)^(ℓ_1…ℓ_n)→ K,     c_(ℓ_1…ℓ_n)^(k_1… k_n):W_(ℓ_1…ℓ_n)^(k_1… k_n)× W_(k_1… k_n)^(ℓ_1…ℓ_n)→ K

by making them on V× V_j'^i' (resp. on W× W_j'^i'), where ,i',,j'∈ I_∞' (resp. ,i',,j'∈ I_∞” with i_1,i_1'∞) equal to b (resp. c1mu) when i'=, j'= and zero otherwise. 

Define p^k_1+1-semilinear isomorphisms 

    ν_(ℓ_1…ℓ_n)^(k_1k_2… k_n+1):V_(ℓ_1…ℓ_n)^(k_1k_2… k_n+1)→ W_(k_2… k_n+1)^(k_1ℓ_1…ℓ_n)

assuming them to be equal to ν on V for ∈ I_∞'.

Define the flags _(ℓ_1…ℓ_n)^(k_1… k_n) and _(ℓ_1…ℓ_n)^(k_1… k_nk_n+1) as follows. The space V_(ℓ_1…ℓ_n)^(k_1… k_n)(q) is the sum of all spaces V with ∈ I_∞' such that 

    i_1=k_1,…,i_n=k_n,     j_1=ℓ_1,…,j_n=ℓ_n,
    
    i_n+1<q  for n even,     i_n+1⩾ q  for n odd.

The space V_(ℓ_1…ℓ_n)^(k_1… k_nk_n+1)(q) is the sum of all V with ∈ I_∞' such that

    i_1=k_1,…,i_n+1=k_n+1,     j_1=ℓ_1,…,j_n=ℓ_n,
    
    j_n+1⩾ q  for n even,     j_n+1<q  for n odd.

The values of q are determined in each case by condition (<ref>) or (<ref>).

The flags _(ℓ_1…ℓ_n)^(k_1… k_n) and _(ℓ_1…ℓ_n)^(k_1… k_nk_n+1) are similarly defined. Then for ∈ I_∞' (resp. ∈ I_∞” with i_1∞) we have

Φ_i_n+1_(j_1…j_n)^(i_1…i_n)   ≅V_(j_1…j_n)^(i_1…i_ni_n+1),       Φ_j_n+1_(j_1…j_n)^(i_1…i_ni_n+1)   ≅V_(j_1…j_nj_n+1)^(i_1…i_ni_n+1),


Φ_i_n+1_(j_1…j_n)^(i_1…i_n)   ≅W_(j_1…j_n)^(i_1…i_ni_n+1),       Φ_j_n+1_(j_1…j_n)^(i_1…i_ni_n+1)   ≅W_(j_1…j_nj_n+1)^(i_1…i_ni_n+1).

From the constructed ingredients we build up A_n∈_n and B_n∈_n for each n⩾ 0. Then B_n≅__n^_nA_n and A_n+1≅__n+1^_nB_n. For n sufficiently large, the objects A_n are primitive and the corresponding objects of  are equivalent to C. Therefore, the objects A_0 and C are equivalent.

An object A∈ is decomposable if and only if A is decomposable.


Proof.
If A is decomposable, then all objects __n^A are decomposable, which implies decomposability of A.

Conversely, suppose A is equivalent to the orthogonal sum C⊕ C,  where C,  C 0. By Lemma <ref> C and C are equivalent, respectively, to A and A for some A,  A∈. Then A is equivalent to ( A⊕ A). By Lemma <ref> A is equivalent to A⊕ A, i.e., A is decomposable.

Set 

:   =I'_∞×I'_∞,
       :
   ={()∈I”_∞×I”_∞|i_1∞},

_σ:   ={()∈|i_2∞},
       _τ:
   ={()∈|j_1∞}.

Define[Note that τ and τ' are involutive permutations, ρ is a bijection, and σ an injection.]

    τ:→,    τ':_τ→_τ,    ρ:→,    σ:_σ→

by the formulas 

    τ((i_1i_2…),(j_1j_2…))   =((j_1j_2…),(i_1i_2…)),
    τ'((i_1i_2…),(j_1j_2…))   =((j_1j_2…),(i_1i_2…)),
    ρ((i_1i_2i_3…),(j_1j_2…))   =((i_1j_1j_2…),(i_2i_3…)),
    σ((i_1i_2i_3…),(j_1j_2j_3…))   =((i_3i_4i_5…),(i_2i_1j_1j_2…)).

Thus, _σ=ρ^-1(_τ) and σ=τρ^-1τ'ρ.  Since τ^2=𝕀 and τ^' 2=𝕀, we have

    στσ|_ _σ=τ|_ _σ.

Consider  as a quiver with an arrow going from point P∈ to point Q∈ if and only if P∈_σ and Q=σ P. Formula (<ref>) implies that τ is an anti-automorphism of the quiver , i.e., τ reverses arrows. The connected components of the quiver  are linear or cyclic directed chains without branching. Define α:→ by the formula

    α((i_1i_2…  ),(j_1j_2…  ))=i_1+1.


Consider a category whose objects will be called symplectic representations or, for short, simply representations of the quiver . They are described as collections

    ((V_P)_P∈, (b_P)_P∈, (h_P)_P∈_σ),

where the V_P are finite-dimensional vector spaces among which there are finitely many non-zero ones,

    b_P:V_P× V_→ K

are non-degenerate pairings,

    h_P:V_P→ V_

are p^α(P)-α(τσ P)-semilinear isomorphisms such that 

    b_P(u,v)=-b_(v,u)  for all  u∈ V_P,  v∈ V_,
    
    b_P(u,u)=0  for all  u∈ V_P,  if  τ P=P,
    
    b_P(u,h_1mu v)^ p^α(P)=b_(h_P1mu u,v)^ p^α(τσ P)  for all  P∈_σ, u∈ V_P, v∈ V_,
    b_P(u,h_P1mu u)=0  for all  u∈ V_P,  if  P∈_σ,  σ P=τ P.

The support of the representation (<ref>) is the set {P∈| V_P 0}. The notions of equivalence of representations, orthogonal sums of representations, decomposable and indecomposable representations are defined in a natural way. 

The category  is equivalent to the category of symplectic representations of the quiver . The orthogonal sum decompositions in the two categories  correspond to each other.


Proof. Given an object (<ref>), we write

    V_P=V,   b_P=b,  ν_P=ν  for  P=()∈,
    
    W_Q=W  for  Q=()∈,    and     c_Q=c  for  Q=()∈_τ.

For P∈_σ, define h_P as the composite of isomorphisms

    h_P: V_Pν_P⟶W_ρ P⟶(W_τ'ρ P)^*=(W_ρ1mu)^*⟶(V_)^*⟶ V_.

The second mapping here comes from the pairing c_ρ P. The penultimate mapping assigns to a linear form ψ∈(W_ρ1mu)^* the linear form φ∈(V_)^* given by the formula

    φ(v)=ψ(ν_(v))^ p^-α(τσ P).

It is therefore p^-α(τσ P)-semilinear. The last mapping comes from the pairing b_. As a result, h_P is p^α(P)-α(τσ P)-semilinear. The definition of h_P amounts to the identity 

    b_σ P(h_P1mu u,v)=c_ρ P(ν_P1mu u,ν_1mu v)^ p^-α(τσ P)  for all  u∈ V_P,  v∈ V_.

Let us replace here P with τσ P. Taking into account equality (<ref>) we get

    b_(h_1mu v,u)=c_τ'ρ P(ν_1mu v,ν_P1mu u)^ p^-α(P),

which can be rewritten as

    b_P(u,h_1mu v)=c_ρ P(ν_P1mu u,ν_1mu v)^ p^-α(P).

Comparing this with (<ref>) we get (<ref>). If σ P=τ P, then (<ref>) with v=u yields (<ref>). Thus,  a representation (<ref>) is constructed.

On the other hand, from the representation (<ref>) we can recover the object (<ref>) as follows. For P∈, select arbitrarily a space W_ρ P of the same dimension as V_P and a p^α(P)-semilinear isomorphism ν_P:V_P→ W_ρ P. Now define pairings c_ρ P by formula (<ref>).

Denote by  the subset of pairs ()∈ such that  simultaneously either contain ∞, or do not contain it and in this case are periodic.


The support of any representation (<ref>) is contained in .


Proof.
Since the set {P∈| V_P 0} is finite, for each Q=() taken from this set there exists n>0 such that either σ^nQ and σ^-nQ are not defined, or σ^nQ=Q. By the definition of σ, in the first case  and  contain ∞, while in the second case they do not contain it and are periodic.

A set of the form X∪τ X, where X is a connected component, will 
be called a τ-connected component of the quiver . 
Denote by  the set of pairs (), where  are sequences of 
positive integers both either finite of the same length, or infinite periodic. 
Consider an equivalence relation on  generated by relations 

    ((k_1… k_n),(ℓ_1…ℓ_n))∼((ℓ_n…ℓ_1),(k_n… k_1)),
    
    ((k_1k_2…  ),(ℓ_1ℓ_2…  ))∼((k_2k_3…  ),(ℓ_2ℓ_3…  )),
    
    ((k_1k_2…  ),(ℓ_1ℓ_2…  ))∼((ℓ_tℓ_t-1…ℓ_1ℓ_tℓ_t-1…  ),(k_tk_t-1… k_1k_t…)),

where t is the common period of the two sequences.


The τ-connected components of  contained in  are in a one-to-one correspondence with the points of the set /∼.


Proof. The components of interest to us are of the form

    -220pt[0pt][0pt]
        < g r a p h i c s >
    -10pt
        < g r a p h i c s >
    
    -10pt
        < g r a p h i c s >
    
    -30pt
        < g r a p h i c s >
    
    -30pt
        < g r a p h i c s >

where the reflection in the dashed line corresponds to the action of the anti-automorphism τ.

To components of the form (<ref>), (<ref>) we assign the class of the pair

    ((α(P_1),…,α(P_n)), (α(τ P_1),…,α(τ P_n)))

in /∼, and to components of the form (<ref>), (<ref>) the class of

    ((α(P_1),…,α(P_t),α(P_1),…,α(P_t),…), (α(τ P_1),…,α(τ P_t),α(τ P_1),…,α(τ P_t),…)).

If 

    P_1=((i_1i_2…  ),(j_1j_2…  )),

then

    σ^rP_1   =((i_2r+1i_2r+2…  ),(i_2r… i_1j_1j_2…  )),
    σ^-rP_1   =((j_2r… j_1i_1i_2…  ),(j_2r+1j_2r+2…  )).

This implies that the class in /∼ represented by ((k_1… k_n),(ℓ_1…ℓ_n))∈ corresponds to the unique τ-connected component containing the point[Subtracting 1 from k_1,ℓ_1,… is required to conform with the previously used indexing scheme,  where we have α()=i_1+1. This corrects the formulas for P_1 in the original manuscript.]

    P_1=((k_1-1,ℓ_2-1,k_2-1,…,ℓ_n-1,k_n-1,∞…  ),
     (ℓ_1-1,∞…  )),

and the class represented by ((k_1k_2…  ),(ℓ_1ℓ_2…  ))∈ corresponds to the component containing the point

    P_1=((k_1-1,ℓ_2-1,k_2-1,…,ℓ_t-1,k_t-1,ℓ_1-1,k_1-1,ℓ_2-1,k_2-1,…  ),
    
    (ℓ_1-1,k_t-1,ℓ_t-1,… k_2-1,ℓ_2-1,k_1-1,ℓ_1-1,k_t-1,ℓ_t-1,…)),

where t is the common period of the sequences .

Clearly, any representation (<ref>) decomposes uniquely as the orthogonal sum of representations with the support of each summand being a τ-connected component. Let us examine one by one representations whose support is a component of some type.

First, consider the component (<ref>). Here a representation is determined, up to an equivalence, by the space V_P_1, and any decomposition of the space into a direct sum leads to the decomposition of the representation into an orthogonal sum. In this case, there exists a unique indecomposable representation corresponding to a one-dimensional space V_P_1.

Consider representations with support (<ref>). Define a non-degenerate bilinear form on V_P_1 setting

    (u,v)=b_P_1(u,h_P_n-1∘…∘ h_P_1(v)).

This form is antisymmetric since

    (u,u)=b_P_s(h_P_s-1⋯ h_P_1(u),h_P_sh_P_s-1⋯ h_P_1(u))^ p^α(P_n-1)+…+α(P_s+1)-α(P_1)-…-α(P_s-1)=0

by (<ref>), (<ref>), when n=2s is even, and 

    (u,u)=b_P_s(h_P_s-1⋯ h_P_1(u),h_P_s-1⋯ h_P_1(u))^ p^α(P_n-1)+…+α(P_s)-α(P_1)-…-α(P_s-1)=0

by (<ref>), (<ref>), when n=2s-1 is odd.

The non-degenerate symplectic space V_P_1 determines the representation up to an equivalence, and any decomposition of V_P_1 into a direct sum of orthogonal subspaces leads to the decomposition of the representation. We have again a unique indecomposable representation corresponding to a two-dimensional symplectic space.

Consider representations with support (<ref>). Let

    k_1=α(P_1),…,k_t=α(P_t),k_t+1=α(P_1),…,ℓ_1=α(Q_1),…,ℓ_t=α(Q_t),ℓ_t+1=α(Q_1),…

So the component (<ref>) corresponds to the class of () in /∼.

The composite mapping h=h_P_ 1mu t∘ h_P_ 1mu t-1∘…∘ h_P_ 1mu 1 is a non-singular p^k_1+…+k_t-ℓ_1-…-ℓ_t-semilinear endomorphism of the space V_P_1.

The orthogonal decompositions of the representation correspond to the decompositions of V_P_1 into direct sums of h-invariant subspaces. Two representations are equivalent if and only if the corresponding pairs (V_P_1,h) and ( V_P_1, h) are equivalent in the sense that there exists a linear isomorphism φ:V_P_1→ V_P_1 such that h=φ hφ^-1.

In <cit.>, it is demonstrated that the theory of semilinear endomorphisms reduces to the representation theory of an associative ring. From the Krull-Remak-Schmidt theorem[This is one of basic results in the theory of rings and modules. More generally, the theorem can be formulated for groups with a set of endomorphisms (e.g., <cit.>). Application of this theorem in our case is justified by the fact that only finite-dimensional representations are considered.] it follows therefore that the decomposition of the representation into an orthogonal sum of indecomposable representations is unique up to a permutation of summands and their equivalence.

Finally, let us study representations with support (<ref>). Let τ P_1=P_s. On the space V_P_1 define a bilinear form 

    (u,v)=b_P_1(u, h_P_s-1∘ h_P_s-2∘…∘ h_P_1(v))

and a non-singular linear endomorphism h=h_P_t∘ h_P_t-1∘…∘ h_P_1. The form is non-degenerate thanks to non-degeneracy of the pairing b_P_1. In view of (<ref>), (<ref>), (<ref>) it is also antisymmetric and 

    (u,hu)=0  for all  u∈ V_P_1.

In this case everything is reduced to the study of the pairs (V_P_1,h), where V_P_1 is a non-degenerate symplectic space and h is a non-singular linear endomorphism satisfying condition (<ref>).


1) Let V be a non-degenerate symplectic space, f a non-singular linear endomorphism of V such that (u,fu)=0 for all u∈ V. Then V can be represented as a direct sum U⊕ U' of two isotropic f-invariant subspaces.

2) If ( V, f) is another such pair and V= U⊕ U' is a similar decomposition, then the pairs (V,f) and ( V, f) are equivalent if and only if the endomorphisms f|_U and f|_ U are equivalent (cf. Exercise 15b) in  5 of Chapter 9 in <cit.>).


Proof. 1) We will prove by induction on V. Let U_1 be an invariant indecomposable with respect to f subspace of maximal dimension. Since U_1 is indecomposable, it is spanned by {v,fv,f^2 v,…} for a certain v∈ U_1. In particular, U_1 is isotropic.

Further, there exists a smallest non-zero f-invariant subspace U_1^min⊆ U_1. Let U_1' be an arbitrary invariant indecomposable with respect to f subspace such that (U_1^min,U_1') 0. Since U_1^min⊈U_1∩(U_1')^⊥, we have U_1∩ U_1^'⊥=0.

The pairing U_1× U_1'→ K has no left kernel, and since U_1'⩽ U_1, it has no right kernel either. Thanks to indecomposability U_1' is isotropic. Therefore, the restriction of the form to U_1⊕ U_1' is non-degenerate. It remains to apply the induction hypothesis to the orthogonal complement of the subspace U_1⊕ U_1'.

2) Let V=U⊕ U' be any decomposition with the required properties. Let us represent U as a direct sum of indecomposable with respect to f subspaces U_1,…,U_n. Then U' can be expressed as a direct sum of indecomposable with respect to f subspaces U_1',…,U_n' so that (U_i,U_j')=0 for i j, and the restriction of the form to U_i× U_i' is a non-degenerate pairing for each i. The minimal polynomials of the endomorphisms f|_U_i and f|_U_i' coincide thanks to the identity (u,fv)=(fu,v). Therefore, f|_U_i and f|_U_i' are equivalent for each i. Hence, so are the endomorphisms f|_U and f|_U'. 

If the pairs (V,f) and ( V, f) are equivalent, then the endomorphisms f and f are equivalent. By the above, the endomorphisms f|_U and f|_U are also equivalent. Conversely, if f|_U and f|_U are equivalent, then the identity (u,fv)=(fu,v) implies the equivalence of (V,f) and ( V, f).

Thus, the equivalence classes of representations of the quiver  with support (<ref>) correspond to the equivalence classes of non-degenerate linear endomorphisms. In particular, the decomposition of such a representation into an orthogonal sum of indecomposables is unique.

Summarizing what we have said above, we conclude that an arbitrary representation (<ref>) decomposes into an orthogonal sum of indecomposables uniquely, up to a permutation of summands and their equivalence.

By Lemmas <ref>, <ref> the equivalence classes of objects of  are in a one-to-one correspondence with the equivalence classes of representations of the quiver . Moreover, it follows from Lemmas <ref> and <ref> that the orthogonal sum decompositions are respected under this correspondence. This implies uniqueness of the decomposition of any object of the category  as an orthogonal sum of indecomposable objects, up to a permutation of summands and their equivalence.

Let us find indecomposable objects of the category  explicitly.

For any object (<ref>), there exist bases e_1,…,e_N in V and e_1',…,e_N' in W compatible with the flags  and , respectively, such that 

    m_i=min{q| e_i∈ V(q)}=min{q| e_i'∈ W(q)}

and the mapping ν^m_i-1 transforms the class e_i of the element e_i in Φ_m_i-1 into the class e_i' of the element e'_i in Φ_m_i-1. Such bases will be called compatible.

The collection m_1,…,m_N carries complete information about the flags  and  and the isomorphisms ν^k:

    V(q)=⟨ e_i| m_i⩽ q⟩_K,     W(q)=⟨ e_i'| m_i⩽ q⟩_K,
    ν^ke_i=e_i'  for all i such that m_i-1=k.

Thus,  any object (<ref>) can be described by the collection m_1,…,m_N and the matrices of the forms b, c in compatible bases. 

Let us show how to select compatible bases for the object corresponding to a representation (<ref>). We will use explicit constructions given in the proofs of Lemmas <ref> and <ref>. We have

    V=⊕_P∈ V_P,     W=⊕_P∈ W_ρ P.

Choose bases in all spaces V_P with P∈ so that for τ P P the bases in V_P and in V_τ P are dual to each other with respect to b_P, whereas for τ P=P the matrix of the form b_P in the chosen basis of V_P is in the normal form.

In V consider the basis e_1,…,e_N obtained as the union of the bases of the spaces V_P. Then m_i=α(P) if e_i∈ V_P. Let us order the basis of V so that the matrix of the form b were equal to (here E is the unit matrix of the corresponding size) 

    ([        0        E;       -E 0 to12pt ]).

In each space W_ρ P with P∈ take the basis consisting of the vectors {e_i'=ν_P1mu e_i| e_i∈ V_P}. The union of these bases gives a basis e_1',…,e_N' of W compatible with the basis of V. The matrix of the form c is obtained from (<ref>).

Let us specify the choice of a basis in V for the case of an indecomposable representation. Assuming a basis of W to be constructed as above, let us write down the quantities that determine the object (<ref>). 

Consider the representation with support (<ref>) corresponding to a 1-dimensional space V_P_1, or with support (<ref>) corresponding to a 2-dimensional symplectic space V_P_1. Select e_1∈ V_P_1 and e_n+1∈ V_τ P_1 so that b_P_1(e_1,e_n+1)=1. Set

    e_2:=h_P_1e_1,…,   e_n:=h_P_n-1e_n-1,   e_n+2:=h_τ P_2^-1e_n+1,…,   e_2n:=h_τ P_n^-1e_2n-1.

Then

    m_1=α(P_1),…,m_n=α(P_n), m_n+1=α(τ P_1),…,m_2n=α(τ P_n)

and the matrix of the form c acquires the shape[The superscript ⊤ is used to denote transposed matrices.] 

    ([           0 _n(0) to7pt;    -_n(0)^⊤    0 to14pt ]),

where _n(0) is the Jordan cell with eigenvalue 0.  

Consider a representation with support (<ref>) or (<ref>). Set h:=h_P_t∘ h_P_t-1∘…∘  h_P_1. In the case of support (<ref>) we use the decomposition V_P_1=U_P_1⊕ U_P_1' with the properties described in Lemma <ref>. Set

    U_P_2:=h_P_1U_P_1,   U_P_2':=h_P_1U_P-1',…,   U_P_t:=h_P_t-1U_P_t-1,   U_P_t':=h_P_t-1U_P_t-1'.

Then b_P_i(U_P_i,U_τ P_i)=0, b_P_i(U_P_i',U_τ P_i')=0, and the restriction of b_P_i to U_P_i× U'_τ P_i gives a non-degenerate pairing. Take any basis e_1,…,e_n of the space V_P_1 (resp. U_P_1') in the case of component (<ref>) (resp. (<ref>)) and the dual with respect to b_P_1 basis e_tn+1,…,e_tn+n of the space V_τ P_1 (resp. U'_τ P_1). Set

    e_n+1=h_P_1e_1,…,e_2n=h_P_1e_n,…,e_tn-n+1=h_P_t-1e_tn-2n+1,…,e_tn=h_P_t-1e_tn-n,
    
    e_tn+n+1=h_τ P_2^-1e_tn+1,…,e_tn+2n=h_τ P_2^-1e_tn+n,…,e_2tn-n+1=h_τ P_t^-1e_2tn-2n+1,…,e_2tn=h_τ P_t^-1e_2tn-n.

Then

    m_1=…=m_n=α(P_1),  …,   m_tn-n+1=…=m_tn=α(P_t),
    
    m_tn+1=…=m_tn+n=α(τ P_1),  …,   m_2tn-n+1=…=m_2tn=α(τ P_t).

Let (ξ_ij) be the matrix of the endomorphism h (resp. h|_U_P_1) in the basis e_1,…,e_n in the case of representation with support (<ref>) (resp. (<ref>)). In both cases the matrix of the form c in the basis 
e_1',…,e_N', where N=2tn, of the space W obtained from the basis e_1,…,e_N of V as explained earlier is

    ([                           E;      4c|-4pt0             ⋱;                           E;                       to6pt;               -^1mu⊤ to14pt;            -E;                           ⋱        4c6pt0;                          -E               ])

where

    =([ ξ_11^p^ℓ_1…ξ_t1^p^ℓ_1;                     1; ξ_1t^p^ℓ_1…ξ_tt^p^ℓ_1 ]),  ℓ_1=α(τ P_1).

The obtained object of the category  is indecomposable if the matrix 
(ξ_ij) defines an indecomposable p^α(P_1)+…+α(P_t)-α(τ P_1)-…α(τ P_t)-semilinear endomorphism.

In the case of an algebraically closed field we can specify further. The indecomposable linear endomorphisms are precisely the Jordan blocks. Therefore, in (<ref>) we should take

    =_n(λ)  with  λ 0  when  α(P_1)+…+α(P_t)=α(τ P_1)+…α(τ P_t).

On the other hand, for r 0 any non-degenerate p^r-semilinear endomorphism of a finite-dimensional vector space acts as the identity transformation on a suitable basis. For r=1, this is proved in <cit.>. In the general case the proof goes through without changes. In particular, the space of indecomposable such an endomorphism is 1-dimensional. Therefore, we should replace (<ref>) with (1s and -1s occupy main diagonals of respective blocks)

    ([                 1; 4c|-4pt0        ⋱;                 1;          1 to 5pt;                -1  to 12pt;       -1;                 ⋱   4c6pt0;                -1          ])

when α(P_1)+…+α(P_t)α(τ P_1)+…+α(τ P_t).



§ NORMAL SHAPE OF THE THE 1ST-TYPE SYMPLECTIC FORMS


Consider all possible triples (E,,), where E is a space,  a flag, ∈^2(E,) a symplectic form. Triples (E,,) and (E',',') will be called equivalent if there exists an isomorphism σ:(E,)→(E',') which preserves the divided powers and transforms  to '.  Preservation of divided powers under σ means that for each f∈(E,) we have f^(r)∈(E,) if and only if (σ f)^(r)∈(E','), and if f^(r)∈(E,), then σ f^(r)=(σ f)^(r). (Note that such an isomorphism extends to an isomorphism of complexes (E,)→(E',').)

Suppose that E=E'⊕ E” is a decomposition such that 

    E_k=(E'∩ E_k)+(E”∩ E_k)  for all  k.

Define flags ' (resp. ”) in E' (resp. E”) by setting E_k'=E'∩ E_k and E”_k=E”∩ E_k. There are canonical embeddings (E',')→(E,) and (E”,”)→(E,). 

A form ∈^1 will be called decomposable if it is conjugate under G() to a form which can be written as '+” for some '∈^1' and ”∈^2”.

To every triple (E,,) we assign an object (<ref>) as follows. 
Set

    V=E^*    and     W=H^1(())^*.

Define the flags  and  by the formulas

    V(k)   ={v∈ V|⟨ v,E_k⟩=0},
    
    W(k)   ={w∈ W|⟨ w,H^1(())_k⟩=0}

(the flag in H^1(()) has been defined in <ref>). The isomorphisms of Proposition <ref> induce p^k+1-semilinear isomorphisms 

    V(k+1)/V(k)≅(E_k/E_k+1)^*→(H^1(())_k/H^1(())_k+1)^*≅ W(k+1)/W(k)

which we take for ν^k. The mapping (<ref>) assigns to  bivectors in  and ⋀^2H^1(()) which we identify with forms b and c, respectively.


Symplectic forms ∈^2(E,) and ∈^2( E,) are equivalent if and only if so are the objects A and A of the category  corresponding to them.

The form  is indecomposable if and only if the corresponding object A is indecomposable.


Proof. Suppose that A and A are equivalent. Then the identically indexed factors of the flags  and , and therefore also such factors of the flags  and , have the same dimension. Hence, we may assume that E=E and =. Then A being equivalent to A means precisely that the images of  and  in the set ×⋀^2H^1(()) are conjugate under the group T (see the definition at the end of <ref>). Since (<ref>) is a bijection, the forms  and  are conjugate under G().

Suppose that A≅ A'⊕ A” is a decomposable object. The objects A' and A” correspond to some symplectic forms '∈^2(E',') and ”∈^2(E”,”). On E'⊕ E” define a flag '⊕” whose spaces are E_k'⊕ E”_k. Then '+”∈^2(E'⊕ E”,'⊕”) is a decomposable symplectic form, and the corresponding to it object in  is equivalent to A'⊕ A”, i.e., to A. By the preceding part  is equivalent to '+”, i.e., is decomposable.

Using the classification of objects in the category  given in <ref> let us formulate our final results. Recall that  stands for the set of pairs (),  where  and  are sequences of positive integers, both of which are either finite, or infinite periodic. The equivalence relation considered on  is generated by (<ref>)–(<ref>).


For each symplectic form of the 1st type the equivalent form represented as the sum of indecomposable symplectic forms in disjoint groups of indeterminates[By indeterminates x_1,…,x_n we mean here elements of an -compatible basis of the space E. As explained in <ref>, the flag  determines the height m_i of each x_i. Since  is not fixed in this section, a precise description of a symplectic form should also include the values of the heights m_1,…,m_n, from which the flag can be reconstructed.] is unique up to a permutation of summands and their equivalence.


[This is a new formulation which should be, hopefully, better understandable than the original formulation.]

The indecomposable symplectic forms of the 1st type split up into disjoint classes which are in a bijective correspondence with the elements of the set /∼.

Any indecomposable 1st-type symplectic form is equivalent to a form

    ∑_i<j(a_ij+b_ijx_i^(p^m_i-1)x_j^(p^m_j-1)dx_i∧ dx_j),

where a_ij,b_ij∈ K, the matrix (a_ij) is as presented in (<ref>), whereas the shape of the matrix (b_ij) is determined by the class containing the given form.

Let ()∈ be a representative of an element of the set /∼. The corresponding class consists of equivalent symplectic forms with the matrix (b_ij) in their normal shape given by (<ref>), if the sequences 

    =(k_1,…,k_n)  and   =(ℓ_1,…,ℓ_n)

are finite. Moreover, the heights of the indeterminates x_1,…,x_2n in this case are

    m_1=k_1,…,m_n=k_n, m_n+1=ℓ_1,…,m_2n=ℓ_n.


If the sequences ,  are infinite and t is their least common period, then the matrices (b_ij) of the forms in the class corresponding to the pair () are as in (<ref>) with the blocks arising from a splitting of the indeterminates x_1,x_2,… into 2t groups of equal cardinality. 

All indeterminates in the first of these groups have height k_1,… , indeterminates in the t-th group have height k_t, indeterminates in the (t+1)st group have height ℓ_1,… , and indeterminates  in the last 2t-th group have height ℓ_t. Each matrix of shape (<ref>) depends on a matrix (ξ_ij) of an indecomposable p^k_1+…+k_t-ℓ_1-…-ℓ_t-semilinear endomorphism. 

Two symplectic forms in the same class are equivalent if and only if so are the corresponding semilinear endomorphisms.

If K is algebraically closed, then in the second case, i.e., if  are infinite periodic, the matrix (b_ij) is the matrix (<ref>),  where  is a Jordan cell with nonzero eigenvalue if k_1+…+k_t=ℓ_1+…+ℓ_t, or the matrix (<ref>) of size 2t× 2t if k_1+…+k_tℓ_1+…+ℓ_t.


99
[Bou]1
Bourbaki N., Algèbre, Chapitre 9, Hermann, 1959.

[H]2
Hodge W.V.D., Pedoe D., Methods of Algebraic Geometry, Volume 1, Cambridge Univ. Press, 1953.

[Jr]3
Jacobson N., The Theory of Rings, AMS, 1943.

[Jlie]4
Jacobson N., Lie Algebras, Interscience Publishers, 1962.

[KfiD]KfiD
Kac V.G., Description of filtered Lie algebras with which graded Lie algebras of Cartan type are associated (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 800–834. English translation in Math. USSR Izv. 8 (1974) 801–835.

[KSh]KSh
Kostrikin, A. I., Shafarevich, I.R., Graded Lie algebras of finite characteristic (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969) 251-322. English translation in Math. USSR Izv. 3 (1969) 237–304.

[Kry]kry
Kryluk Ya. S., Algebras of Cartan type: representations and prolongations (in Russian), Ph.D thesis, Moscow State Univ., 1978.

[KuKi]5
Kuznetsov, M. I.,  Kirillov S.A., Hamiltonian differential forms over the algebra of truncated polynomials, Uspekhi Mat. Nauk 41:2 (1986) 197–198. English translation in Russian Math. Surveys, 41:2 (1986) 205–206.

[Sk]Sk
Skryabin, S.M. , Classification of hamiltonian forms over divided power algebras (in Russian), Mat. Sbornik 181 (1990) 114–133. English translation in Math. USSR Sbornik 69 (1991) 121–141.

[St]St
Strade H., Simple Lie Algebras over Fields of Positive Characteristic, I. Structure Theory, de Gruyter, 2004.

[Tyu]Tyu
Tyurin S.A., Classification of deformations of the special Lie algebra of Cartan type (in Russian), Mat. Zametki 24 (1978) 847–857. English translation in Math. Notes 24 (1978) 948–954.

[Wcl]10 
Wilson R. L., Classification of generalized Witt algebras over algebraically closed fields, Trans. Amer. Math. Soc. 153 (1971) 191–210.

[Waut]11 
Wilson R. L., Automorphisms of graded Lie algebras of Cartan type, Comm. Algebra 3 (1975) 607–608.

[Ws]12 
Wilson R. L., Simple Lie algebras of type S, J. Algebra, 62 (1980) 292–298.


