Gerstenhaber structure on Hochschild cohomology of toupie algebras
Dalia Artenstein Marcelo Lanzilotta Andrea Solotar


How Developers Iterate on Machine Learning Workflows
    Doris Xin, Litian Ma, Shuchen Song, Aditya Parameswaran
    
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    We study homological properties of a family of algebras called toupie algebras. Our main objective is to obtain the 
    Gerstenhaber structure of their Hochschild cohomology, with the purpose of describing the Lie algebra structure of the first 
    Hochschild cohomology space, together with the Lie module structure of the whole Hochschild cohomology.   
      






Keywords: Hochschild cohomology, Gerstenhaber algebra.






§ INTRODUCTION



 In this article we study homological properties of toupie algebras, first defined in <cit.>.
Toupie algebras combine features of canonical algebras with monomial algebras. Canonical algebras were introduced by Ringel 
in <cit.>, see also <cit.> for historical references about canonical algebras.

An algebra is toupie if it is a quotient of the path algebra of a finite quiver Q which
has a source 0, a sink ω and branches going from 0 to ω by an ideal I⊆ Q_≥ 2 generated by a set
containing two types of relations: monomial ones, which involve arrows of one branch each, and linear combinations of branches.

Canonical algebras are part of a more general class, the concealed-canonical algebras, see <cit.>. Since one of the properties 
distinguishing canonical algebras within the class of concealed-canonical algebras is that their quiver has only one sink and only 
one source <cit.>, we
conclude that toupie algebras and concealed-canonical algebras only share the subfamily of canonical algebras.


Almost all toupie algebras are of wild representation type, see <cit.>. Toupie algebras are also special multiserial algebras, 
see <cit.> for the definition, which are usually of wild representation type too. As a consequence, modules over toupie algebras 
are multiserial, that is non necessarily  direct finite sums of uniserial modules. The Hochschild cohomology of special multiserial 
algebras is still unknown except for some particular examples.

The Hochschild cohomology ^*(A) of an algebra A, together with its associative algebra structure given by the cup product and 
its Gerstenhaber algebra structure,
is a derived invariant. Even having an explicit description of ^*(A), the cup product and the Gerstenhaber bracket are 
not easy to compute. Bustamente proved in <cit.> that the cup product of ^*(A) is trivial for any triangular quadratic 
string algebra A, and the Gerstenhaber bracket vanishes for elements in cohomological degrees greater than 1 when A is a gentle triangular algebra. These are tame algebras with a particularly easy resolution.
Also, Redondo and Román computed in <cit.> the Gerstenhaber structure of the Hochschild cohomology of a triangular 
string algebra, showing that it is trivial in degrees greater than 1. Subsequently, they  computed in <cit.> the 
Gerstenhaber structure of the Hochschild cohomology of string quadratic algebras. In this case they gave conditions on the quiver 
associated to the string quadratic algebra in order to get non trivial cup product and Gerstenhaber bracket in degrees greater than 1.
The first Hochschild cohomology space of an algebra A is always a Lie algebra with the Gerstenhaber bracket.
Strametz <cit.> described the Lie algebra structure of ^1(A) for A a finite dimensional monomial algebra.
Moreover, Sánchez-Flores made explicit in <cit.> the Lie module structure of higher cohomology spaces 
^n(A) over the Lie algebra ^1(A) when A≃kQ/⟨ Q_2⟩ 
–that is, radical square zero– and Q is either an oriented cycle of length n or a finite quiver with no 
cycles.

In this article we describe the Lie algebra structure of ^1(A), when A is a toupie algebra, as well as the Lie module structure of ^n(A) over ^1(A).
For this, we construct a resolution of A as A-bimodule, using technics of <cit.>. We shall see that the existence of non 
monomial relations will only have an effect in degrees 0, 1, 2 of the resolution but this difference will considerably change 
the Lie structure of the first cohomology group.






Even if the dimensions of the k-vector spaces ^*(A) are already known <cit.>, an explicit computation of 
these is needed for the description of the Gerstenhaber structure.



The contents  of the article are as follows.
In Section <ref> we fix notations and prove some preliminary results.
Section <ref> is devoted to the computation of a k-basis of each Hochschild cohomology space, while 
in Section <ref> we obtain the comparison morphisms between the reduced bar resolution and ours.

In Section <ref> we prove that the Gerstenhaber bracket is zero in degrees greater that 1, and we compute it 
when restricted to ^1(A).

The description of ^1(A) as a Lie algebra is given in Section <ref>, where we find necessary and sufficient conditions for 
it to be abelian and to be semisimple. We describe its centre and we prove that when k=, it has a Lie subalgebra
isomorphic to sl_a(), where a is the number of arrows from 0 to ω in the quiver Q.
Finally, we prove  Theorem <ref>, one of our main theorems.

In Sections <ref> and <ref> we describe the Lie module structure of ^n(A) for n≥ 2. The main results are 
Theorem <ref> and Theorem <ref>. We end the article with an example.








§ PRELIMINARIES


Let k be a field of characteristic zero.

In this section we will recall some definitions, as well as some preliminary results.



 §.§ E-reduced Bar resolution

Given a finite dimensional k-algebra A with radical r such that A=E⊕ r with E a separable 
k-subalgebra E⊂ A, Cibils proved in <cit.> that the following complex is a projective A-bimodule
resolution of A, see also <cit.>

    … A⊗_EA^⊗_E 3⊗_EA  δ_2⟶ A⊗_EA^⊗_E 2⊗_E A  δ_1⟶ A⊗_EA⊗_EA δ_0⟶ A⊗_E A
      δ_-1→ A  → 0

where A=A/E and δ_i is such that 
δ_i(a_0⊗_Ea_1⊗_E…⊗_Ea_i+1⊗_E a_i+2 )=
δ_i(a_0⊗ a_1⊗…⊗ a_i+1⊗ a_i+2).  
From now on we will simply call it δ_i instead of δ_i.
 



 







We will give a contracting homotopy  t={t_i}_i≥ -1 that we are going to use later to define the comparison maps
between this resolution and the minimal one:

    t_i:A⊗_EA^⊗_Ei⊗_EA→ A⊗_EA^⊗_Ei+1⊗_EA

such that δ_i+1∘ t_i+1+t_i∘δ_i=id.
We define t_i(a_0⊗_Ea_1⊗_E…⊗_Ea_i⊗_Ea_i+1)=
1⊗_Ea_0⊗_Ea_1⊗_E…⊗_Ea_i⊗_Ea_i+1.


This projective resolution will be called the E-reduced Bar resolution of A, C_BarE(A) To simplify notation we will 
still denote by  δ_n the differentials of this complex.
It is possible to define the reduced cup product ⌣_red and the reduced Gerstenhaber bracket [  ,  ]_red
in terms of this resolution. Applying the functor  
_A^e(-,A) 
to the E-reduced Bar resolution and considering the 
isomorphism

    F_X:_A^e(A⊗_E X⊗_E A,A)→_E^e(X,A)
 
natural in X given by 

    F_X(φ)(x)=φ(1⊗_E x⊗_E 1),
 
we obtain the following complex:

    0⟶_E^e(E,A) ϵ_0⟶_E^e(A,A)ϵ_1⟶_E^e(A⊗_EA,A)ϵ_2⟶_E^e(A^⊗_E3,A) ⟶…

The reduced cup product in ⊕_i=0^∞_E^e(A^⊗_E i,A) is as follows: 
consider φ∈_E^e(A^⊗_E n,A)  and ϕ∈_E^e(A^⊗_E m,A), 
given a_1, … ,a_n+m∈ A

    φ⌣_redϕ(a_1⊗…⊗a_n+m)=φ(a_1⊗…⊗a_n)ϕ(a_n+1⊗…⊗a_n+m).
 
It is easy to prove that ⌣_red induces a product in Hochschild cohomology that coincides with the usual cup product, 
see remarks 2.3.14 and 2.3.19 of <cit.>.












Analogous modifications hold for the Gerstenhaber bracket, see Appendix B of <cit.>. 






























 §.§ Toupie algebras


We recall some definitions from <cit.>.
A finite quiver is toupie if it has a unique source and a unique sink and any other vertex 
is the source of exactly one arrow and the target of exactly one arrow. The source and the sink will be denoted 0 and 
ω, respectively.

Given a toupie quiver Q and any admissible ideal I⊂kQ, A= kQ/I will be called a 
toupie algebra, the paths from 0 to ω will be called branches. The j-th branch will be denoted 
α^(j). The length of a branch α^(j) –denoted by |α^(j)|– is the number of arrows in it.


We have already observed that canonical algebras are toupie algebras. Besides canonical algebras there are many examples of toupie 
algebras, let us just mention the n-Kronecker algebras.



From now on, A= kQ/I will always denote a toupie algebra. There are four possible kinds of branches in Q and we will 
consider the following order within the branches. First, let Z={α^(1),… ,α^(a)} be the set of arrows from 
0 to ω. The branches α^(a+1), … ,α^(a+l) will be those of length greater than or equal to 2 not 
involved in any relation. Next, α^(a+l+1),… ,α^(a+l+m) will be the branches containing monomial relations and 
finally α^(a+l+m+1), … ,α^(a+l+m+n) will be the branches involved in non
monomial relations.

Denoting by e_x the idempotent of A corresponding to the vertex x, we define:

    D:=dim_ke_0Ae_ω   =a+l+n-#{linearly independent non monomial relations}.



Given a finite set of s equations generating the non monomial relations and having fixed an order of the branches, 
let C=(a_ij)∈k^s× n be the matrix whose rows are the coefficients of each of these equations.
Replacing the given relations by those obtained from the reduced row echelon form of the matrix gives of course the same algebra; 
from now on we will always suppose that this matrix is already reduced.
Every non monomial relation will be of the form:

    ρ_i=α^(k_i)+∑_j>k_ib_ijα^(j).

We will call W_ρ_i=α^(k_i) and f_ρ_i=-∑_j>k_ib_ijα^(j), keeping in mind the idea that 
the word W_ρ_i will be replaced in A by f_ρ_i.
Let ℛ be a minimal set of generators of I containing ρ_i for all i.
The set ℛ will be the disjoint union of 
 ℛ_mon, consisting of monomial relations  and ℛ_nomon={ρ_i}, the set of non monomial relations.

Since we are going to compute the Hochschild cohomology spaces of A, the first thing we need is a useful projective resolution of 
A as A-bimodule; if possible a minimal one.











 §.§ The resolution

We will next recall the definition of n-ambiguity from <cit.>.





 Given n≥2,
 
  
  * the path p ∈ Q is a left n-ambiguity if there exist u_0∈ Q_1 and 
   u_1,…,u_n paths not in I such that
    
      
  * p=u_0u_1⋯ u_n,
      
      
  * for all i, and for any proper left divisor d of u_i+1, the path u_iu_i+1 belongs to I but 
      u_id does not belong to I.
      
    
   
  * the path p∈ Q is a right n-ambiguity if there exist v_0∈ Q_1 and
   v_1,…,v_n paths not in I such that
    
     
  * p=v_n⋯ v_0,
     
  * for all i, and for any proper right divisor d of v_i+1, the path v_i+1v_i belongs to  I but dv_i does not belong to I.
    
 





As we have already mentioned, the subalgebra E= kQ_0 of A is separable over k, so we can compute 
the resolution
relative to E. This resolution will come from a monomial order, see <cit.>. Let us fix an order
in Q_0∪ Q_1. For this, let us draw the toupie quiver Q as follows



    @R=6pt@C=6pt      [lld] [ld]∙[]+<0pt,8pt>*0[dr]@/^4pc/[ddddd] @/^6pc/[ddddd]   
     _e^(1)_1∙[d]    _e^(2)_1∙[d]    ⋯   ∙[d]   
    ⋮   ⋮   ⋯   ⋮                 ⋯
    [d]    [d]       [d]
     _e^(1)_n_1∙@/^-1pc/[drr]    _e^(2)_n_2∙[dr]    ⋯   ∙[ld]    
          ∙[]+<0pt,-8pt>*ω
      











and for each i denote 
{e_j^(i)/

1≤ j≤ n_i} the set of vertices in the 
branch α^(i)
such that
e_j^(i)≠ 0 and e_j^(i)≠ω.
Fix ω=min Q_0, 0=max Q_0 and e_j^(i)<e_k^(l) if l<i or if i=l and k<j. For the arrows, 
fix α_j^(i)<α_k^(l) if l<i or i=l and k<j. Let ≤ be the order in kQ/I which is compatible 
with concatenation and
extends <. Choose S=Mintip (I) as in <cit.> for example. It is then known that I is the two sided
ideal generated by {s-f_s}_s∈ S and that the reduction system ℜ={(s, f_s)}_s∈ S is such that
every path is reduction unique, see for example <cit.>. Explicitly, 
ℜ=ℜ_1∪ℜ_2,
where ℜ_1={(σ_i,0)/ σ_i a monomial relation} and 
ℜ_2={(α^(k_i), Σ_j< k_ib_ijα^(j))/ a+l+m+1≤ k_i≤ a+l+m+n}.

In our situation, it is easy to see that all possible ambiguities arise from ℜ_1, and since they correspond to the 
monomial part, they reduce to 0. We denote by ℬ the basis of A obtained from the classes in kQ/I of the 
irreducible paths.

From now on we will fix the reduction system. We are now ready to construct the resolution.


 Given a path u∈ Q, whenever we write ∑ u^(1)⊗ u^(2)⊗ u^(3)∈ A⊗_EA⊗_EA we are considering 
 the sum of all the possible factorizations of u as product u^(1)u^(2)u^(3)=u. If u^(2) is an arrow we write 
 ∑ u^(1)⊗u^(2)⊗ u^(3). If ρ_i is a non monomial relation, we know that it is of the 
 form α^(k_i)+∑_j> k_i b_ijα^(j), then when we write 
 ∑ [ρ_i]^(1)⊗ [ρ_i]^(2)⊗ [ρ_i]^(3) we refer to 
 ∑ [α^(k_i)]^(1)⊗[α^(k_i)]^(2)⊗ [α^(k_i)]^(3)+
 ∑_j> k_ib_ij∑ [α^(j)]^(1)⊗ [α^(j)]^(2)⊗ [α^(j)]^(3).


In low degrees, we already know that the extended minimal resolution is

    A⊗_Ekℛ⊗_EAd_1→A⊗_EkQ_1⊗_EAd_0→A⊗_EAμ→A→ 0



As usual, μ, d_0 and d_1 are the A-bimodule maps defined by;

    μ(a⊗ b)   =ab  for  a,b∈ A,
    
    d_0(1⊗ v⊗ 1)   =v⊗ 1-1⊗ v   for   v∈ Q_1,
    
    d_1(1⊗σ⊗ 1)   =Σσ^(1)⊗σ^(2)⊗σ^(3)  for  σ∈ℛ.

For n≥ 2, let us
denote P_n=A⊗_Ek𝒜_n⊗_EA, where 𝒜_n is the set of n-ambiguities.


Given v=v_0⋯ v_n=w_n⋯ w_0∈𝒜_n, the differential d_n:P_n→ P_n-1 will be defined 
–using the same notation as before– as follows:
 
    d_n(1⊗ v⊗ 1)={[                                     ∑ v^(1)⊗ v^(2)⊗ v ^(3),  with  v^(2)∈𝒜_n-1 if                                                                   n is odd,;                                         w_n⊗ w_n-1… w_0⊗ 1-1⊗ u_0… u_n-1⊗ u_n,                                                                  if n is even.;                                                                                ].


Let us call C_min(A) the sequence

    …⟶ P_nd_n⟶P_n-1⋯⟶ P_2d_2⟶ A⊗_Ekℛ⊗_EA d_1⟶A⊗_EkQ_1⊗_EA d_0⟶A⊗_EA ⟶ 0



The sequence C_min(A) is a complex of projective A-bimodules.







































For n>2, the only relations involved in P_n are monomial and the proof that d_n∘ d_n+1=0 is in <cit.>.
We have to verify that d_1∘ d_2=0 and d_2∘ d_3=0.
Let us first prove that d_1∘ d_2=0. Given a 2-ambiguity v=v_0v_1v_2=w_2w_1w_0,


    d_1∘ d_2(1⊗ v⊗ 1)   =d_1(w_2⊗ w_1w_0⊗ 1-1⊗ v_0v_1⊗ v_2)
       =Σ w_2(w_1w_0)^(1)⊗(w_1w_0)^(2)⊗ (w_1w_0)^(3)
        - Σ (v_0v_1)^(1)⊗(v_0v_1)^(2)⊗ (v_0v_1)^(3)v_2.

In case there is a term in the first summand not appearing in the second one; then (w_1w_0)^(3) must be a proper divisor of
v_2,  which implies that (w_1w_0)^(2)(w_1w_0)^(3) is a divisor of v_2 and so
w_2(w_1w_0)^(1) would be divisible by v_0v_1, which is 0 in A.
A similar argument is used to prove that every non zero term that appears in the second sum appears in the first one too.
Let us now prove that d_2∘ d_3=0.
Given a 3-ambiguity v=u_0u_1u_2u_3=w_3w_2w_1w_0,

    d_2∘ d_3(v)=d_2(Σ_vv^(1)⊗ v^(2)⊗ v^(3))

with v^(2)∈𝒜_2.
The set of 2-ambiguities appearing in the decomposition of v is finite, so we can order the summands by length of the first term
v^(1). Notice that two different terms cannot have v^(1)'s of the same length: if this happens, then one of the v^(2)'s 
would be
strictly shorter that the other, and this contradicts Lemma 3.1.5 of <cit.>.

Let us order the summands as

    v_1^(1)⊗ v_1^(2)⊗ v_1^(3)< ⋯ < v_n^(1)⊗ v_n^(2)⊗ v_n^(3)

with |v_i^(1)|<|v_i+1^(1)| for i=1,...,n-1.

Given t, i≤ t≤ n, we know after the second item of Lemma 3.1.6 of <cit.> that v_t^(2) and v_t+1^(2) both 
have only two divisors in ℛ, respectively v'_t, v”_t and v'_t+1, v”_t+1. The first part of Lemma 3.1.6 
implies that v'_t+1 equals v”_t, because if not v^(2)_t and v^(2)_t+1 would share a 2-ambiguity, which is 
impossible.

Using the notation v_t^(2)=v'_tu_2,t=w_2,tv'_t+1,
and observing that u_2,1=u_2, since v'_1=u_0u_1, and that w_2,n=w_2 since v'_n+1=w_1w_0, we get

    d_2∘ d_3(1⊗ v⊗ 1)   =-Σ_t=1^nv_t^(1)w_2,t⊗ v'_t+1⊗ v_t^(3)+Σ_i=1^nv_t^(1)⊗ v'_t⊗ u_2,t v_t^(3)
       =-w_3w_2⊗ v'_n+1⊗ 1+1⊗ v'_1⊗ u_2u_3
       =0.

Notice that, since E is separable, E⊗_kE^op is semisimple, thus, all the E-bimodules appearing in the
complex are projective, implying that the A-bimodules P_n are projective.













The complex C_min(A) is a projective resolution of A as A^e-module.












The only thing left to prove is the exactness of the complex, and this follows from <cit.>.
Nevertheless, we shall construct a contracting homotopy {s_i}_i≥ -1 that will be useful in Section <ref>.
 For i=-1 and a∈ A, let

    s_-1(a)=a⊗ e_t(a).



For i≥ 0, let a be an element of the k-basis of irreducible paths ℬ. It is sufficient to define s_i 
on the elements e_s(w)⊗ w⊗ a∈ P_i and extend it to a morphism of left A-modules.
Set

    s_0(e_s(a)⊗ a)=  -∑ a^(1)⊗a^(2)⊗ a^(3),



    s_1(e_s(α)⊗α⊗ a)={[                      e_0⊗ρ_i⊗ e_ω ,                      if  α a=W_ρ_i,;                      e_s(α)⊗σ_j⊗ b, if α a=σ_jb in Q with σ_j monomial,;                                  0,                          otherwise. ].


For i= 2, given r∈ℛ, set 
    s_2(e_s(r)⊗ r ⊗ a)=-∑ (ra)^(1)⊗ (ra)^(2)⊗ (ra)^(3)

 with (ra)^(2)∈𝒜_2 if ra contains a 2-ambiguity and zero otherwise.
Observe that if r=ρ_i is a non monomial relation, then s_2(e_s(r)⊗ r⊗ a)=0.

For i> 2, given w∈𝒜_i-1, 
    s_i(e_s(w)⊗ w ⊗ a)=(-1)^i+1∑ (wa)^(1)⊗ (wa)^(2)⊗ (wa)^(3)

 with (wa)^(2)∈𝒜_i if wa contains an i-ambiguity, and zero otherwise.
 It is straightforward to verify that {s_i}_i≥ -1 is a contracting homotopy.
 



To check that the projective resolution is minimal consider r^e, the Jacobson radical of A^e. This radical is generated by the elements of the form α⊗ e_j and e_j⊗β with  i,j ∈ Q_0 and α, β∈ Q_1 since rad(A⊗ A^op)=A⊗ rad(A^op)+rad(A)⊗ A^op.
It is easy to see that Im(d_i)⊂ P_i-1r^e, so we conclude that C_min(A) is minimal.





§ COMPUTATION OF HOCHSCHILD COHOMOLOGY


In this section we will construct an explicit basis for each cohomology  space of A. The knowledge of such bases will be
useful for the computation of the deformations of toupie algebras and for description of the Gerstenhaber structure.
Applying the functor _A^e(-,A) to the minimal resolution C_min(A)
and using again the canonical isomorphism F_X:_A^e(A⊗_E X⊗_E A, A)→_E^e(X,A), that we will simply denote F,
we obtain the following complex:

    0⟶_E^e(E,A)D_0⟶_E^e( kQ_1,A)D_1⟶_E^e( kℛ,A)D_2⟶_E^e( k𝒜_2,A) ⟶…

where D_i=F∘_A^e(d_i, A)∘ F^-1 for all i.


For any E-bimodule W and any f∈_E^e(W,A), the equalities e_if(w)e_j=f(e_iwe_j),
for every i,j∈ Q_0 and w∈ W mean that w and f(w) share source and target. Using Cibils's notation in <cit.>, 
we will write wf(w) for the morphism
in _E^e(W,A) sending a basis element w of W to f(w) and the other basis elements to zero. Also, given sets 
H and G, we will
denote k(H G) the k-span of {h g}_h∈ H, g∈ G.

Let us denote _0ℬ_ω the subset of all branches belonging to ℬ.



We will next describe explicitly the vector spaces and the differentials appearing in the complex.



Since toupie algebras have no cycles, _E^e(E,A) is just k(Q_0Q_0). 
A careful look shows that the spaces appearing in degrees 1 are 2 are respectively

    _E^e( kQ_1,A)= k(Q_1 Q_1)+ 𝕜(Z (_0ℬ_ω-Z)).


    _E^e( kℛ,A)= k(ℛ_nomon_0ℬ_ω)+ k(_0(ℛ_mon)_ω_0ℬ_ω).

For i≥ 2,

    _E^e( k𝒜_i,A)= k(_0(𝒜_i)_ω_0ℬ_ω).


The differentials are as follows. Given j∈ Q_0 and e_j e_j∈_E^e(E, A),

    D_0(e_j e_j)=∑_α: t(α)=jαα-∑_β: s(β)=jββ.

Given α_r∈ Q_1,

    D_1(α_rα_r)={[ ∑_i:α^(j)∈ρ_iρ_i b_ijα^(j),   if   α_r  is  in  α^(j) and α^(j) is a branch of ρ_i,;                                                      0,                  otherwise. ].

For α^(h)α^(j)∈ Z (_0ℬ_ω-Z),

    D_1(α^(h)α^(j))=0.

Finally, D_i=0 for i≥ 2.

An easy verification shows that, as it is well known when the quiver contains no oriented cycles, ^0(A)=⟨∑_i∈ Q_0e_i e_i⟩.



 §.§ Computation of ^1(A)

To compute ^1(A) we first obtain a basis of Ker D_1 that we modify afterwards so that it contains a basis of 
Im D_0.
For that purpose, using the branches of the quiver of length greater than 1 that do not contain monomial relations, we will 
construct a non oriented  graph Q_ρ as follows.

The graph Q_ρ is the following:

  
  * its vertices are in bijection with the branches in Q that are not arrows and do not contain monomial relations,
  labeled under the name of the corresponding branch,
  
  * there is an edge between the vertices α^(k) and α^(p) if there exists a relation that involves both branches.

We will denote Q_ρ^k, 1≤ k≤ r, the connected components of Q_ρ.


We recall the dimension of ^1(A), computed in <cit.>: 

    dim_k^1(A)=r+m+Da-1.


Let us define the following four subsets of k(Q_1ℬ):



  * C_1={α^i_jα^i_j:α^(i) is a branch which contains monomial relations}.

  * C_2=Z_0ℬ_ω.


  * C_3={α^i_jα^i_j- α^i_0α^i_0:j≠ 0, |α^(i)|>1  and α^(i) does not contain monomial relations}.

  * C_4={∑_α^(i)∈ Q_ρ^kα^i_0α^i_0: k=1…, r}.



The set U=⋃_i=1^4C_i is a basis of Ker D_1.


Since dim_k^1(A)=r+m+Da-1, our knowledge of the dimension of (D_0) implies that dim_k Ker D_1=r+m+Da+# Q_0-2. By construction, U is linearly independent. 
Observe that the disjoint union C_1∪ C_3 is in bijection with the set consisting of all the arrows that do not start 
in 0 and the arrows of the monomial part that start in 0, so # C_1+# C_3 =# Q_0-2+m. 
Since clearly # C_2=Da and # C_4=r we conclude that # U=r+m+Da+# Q_0-2.


Next we will replace U by another basis containing a basis of Im D_0; we will proceed as follows.


  * Replace each element α^i_jα^i_j in C_1 with j≠ 0 by α^i_jα^i_j-
α^i_0α^i_0. 
Let us call C_1' the set consisting of the modified elements and 

    C”_1={α_0^iα_0^i: α^(i) is a branch that contains monomial relations},
 its complement in C_1.

  * 

    

    
  * If a>0, first replace the elements in C_2 of the form α^(i)α^(i) with i≠ 1 by 
    α^(i)α^(i)-α^(1)α^(1) and then replace α^(1)α^(1) by 
    s=∑_α^(i)α^i_0α^i_0 where the sum ranges over all the branches of the quiver.
    
  * If a=0, let us call α^(B) the last branch of the toupie algebra and consider the element of U in 
    C_1⋃ C_4 that contains α_0^Bα_0^B as a summand. Replace this element by 
    s=∑_α^(i)α^i_0α^i_0 where the sum ranges over all the branches of the quiver.
     


Let us call U the set obtained from U in this way. After some direct and tedious computations, it turns out that the 
k-vector spaces generated by U and U coincide.


The set K={s}⋃ C'_1⋃ C_3 is a basis of Im D_0.


We know that dim_kIm D_0=#Q_0-1 because on one hand dim_kKer D_0=1 and on the other 
hand dim_k_E^e(E,A)=#Q_0. Since K⊂ Im D_0, just observe that #C'_1+#C_3=#Q_0-2 and K is linearly independent.

The proof of the following theorem is now immediate.

The classes of the elements of U-K in Ker D_1/Im D_0 form a basis of ^1(A).




 §.§ Computation of ^i(A) for higher degrees

In order to obtain a basis of ^2(A) we will follow the same lines as for ^1(A). Recall that D_2=0 and 
so Ker D_2=_E^e( kℛ,A).

To compute a basis of Im D_1 we will need to define an order ≺' in ℛ_nomon. Every element in 
ℛ_nomon is of the form 

    ρ_i=α^(k_i)+∑_j>k_ib_ijα^(j)

so we say that ρ_i≺' ρ_j if k_i< k_j. This order induces an order on the elements of 
ℛ_nomon_0ℬ_ω as follows,
 
    ρ_iα^(k)≺ρ_jα^(l) if ρ_i≺' ρ_j or ρ_i=ρ_j and  k<l.


For each connected component of Q_ρ associated to a non monomial relation there exists a relation containing 
α^(k_i) with k_i maximum. We will call this relation the last one of the component.


Denote by X the set consisting of all the elements in _0ℬ_ω that are involved in non monomial relations and 
define d=#X.


Consider the following subsets of k(ℛ_0ℬ_ω):




  * B_1={ρ_if_ρ_i:ρ_i∈ℛ_nomon such that ρ_i is not the last one of Q^k_ρ,  with k=1,… ,r},


  * B_2={∑_i: α^(h)∈ρ_i, h≠ k_iρ_i b_ikα^(h) : α^(h)∈ X}.




Observe that # B_1=n-d+l-r and #B_2=d.


The set B_1∪ B_2 is a basis of Im D_1.


It is clear that B_1∪ B_2⊂ Im D_1 since
B_1⊂{D_1(α_0^(i)α_0^(i)):
α^(i)=W_ρ_j for some  j} and
B_2={D_1(α_0^(i)α_0^(i)):α^(i)∈ X}.
We know that dim_kKer D_1=r +m + Da + # Q_0-2 and, on the other hand,
dim_k_E^e( kQ_1,A)=# Q_1+(D-1)a=m+l+n+Da+# Q_0-2.
From the previous computations we deduce that dim_k Im D_1=l+n-r. 

Observe that the union B_1∪ B_2 is disjoint: an element in the intersection of B_1 and B_2 must be of the form 
t=ρ_icα^(k) with α^(k)≠α^(k_i) the only branch except for α^(k_i) in  ρ_i 
and c∈k. Now, since t has only one summand, the branch  α^(k) is not
involved in any other relation. On the other hand  α^(k_i), by definition, is only involved in the relation ρ_i. 
Thus, the branches α^(k) and α^(k_i)= W_ρ_i are the only ones in their connected component of Q_ρ.
So ρ_i = α^(k_i) - cα^(k) is the unique relation  of Q^k_ρ, for some k, and this cannot happen 
due to the definition of B_1. 

Since # B_1=n-d+l-r, #B_2=d and the union is disjoint, it follows that #(B_1∪
B_2)=#B_1+#B_2=l+n-r, so it is sufficient to prove that B_1∪ B_2 generates
Im(D_1). By definition,
D_1(α_0^(i)α_0^(i)) coincides with D_1(α_j^(i)α_j^(i))
for any arrow α_j^(i) in the branch α^(i), also
D_1(∑_i: α^(i)∈ Q_ρ^kα_0^(i)α_0^(i))= 0, for every connected component Q_ρ^k of Q_ρ, so the result is proven.


Again, we will modify the basis B_1∪ B_2, following the next steps.


  * For every element w in B_2 with a summand of the form ρ_j b_jkα^(k) where α^(k) is the first 
branch of f_ρ_j and ρ_j is not the last relation for any connected component of Q_ρ, subtract from w the 
element ρ_jf_ρ_j which belongs to B_1. Let us call B'_2 the set obtained from B_2 in this way. 
The vector spaces spanned by B_1∪ B_2 and by B_1∪ B'_2 are equal. 

  * Consider the order defined in <ref> for ℛ_nomon_0ℬ_ω and the 
matrix associated to the equations corresponding to the rows of B'_2 in its echelon form. 
Reconstruct the set in ℛ_nomon_0ℬ_ω from the echelon form and call it B”_2.

Since ⟨ B_1∪ B”_2⟩=⟨ B_1∪ B'_2⟩ =⟨ B_1∪ B_2⟩, we conclude 
that B_1∪ B”_2 is a basis of Im D_1.

Now we will modify the set 
    N={ρ_iα^(k): ρ_i∈ℛ_nomon and α^(k)∈_0ℬ_ω}∪{σ_iα^(k):σ_i∈_0(ℛ_mon)_ω and α^(k)∈_0ℬ_ω},
 which is clearly a basis of _E^e( kℛ,A), in order to obtain another basis N' of _E^e( kℛ,A) including B_1∪ B”_2.
The procedure is as follows.


  * To have the elements of B_1 in our basis, given ρ_i∈ℛ_nomon and α^(k)∈ f_ρ_i with 
k the minimum of f_ρ_i, replace ρ_iα^(k) by ρ_if_ρ_i.

  * To get the elements of B”_2 in our new basis consider the pivot of every element in B”_2 and replace the element 
having this pivot in N by the corresponding one in B”_2.



The classes of N'-B_1∪ B”_2 in _E^e( kℛ, A)/Im(D_1) form a basis of ^2(A).


The last theorem of this section gives a basis of the Hochschild cohomology of A for higher degrees.


Given i≥ 3, the set B_i={w_jα^(k):α^(k)∈_0ℬ_ω  and  w_j∈_0(𝒜_i-1)_ω} is a basis of ^i(A).


Since D_i=0 for i≥ 3 it follows that ^i(A)=_E^e( k𝒜_i-1,A). The set B_i is a basis of 
this k-vector space so it is also a basis of ^i(A).
















§ COMPARISON MORPHISMS

For the computation of Hochschild cohomology we used a minimal resolution. 
One possible way of computing the Gerstenhaber structure is via comparison morphisms between this minimal resolution and 
the E-reduced Bar resolution in whose existence is guaranteed since both complexes are projective resolutions of 
A as A^e-module.
We will describe them explicitly in the sequel.
Let us call φ:C_min(A)→ C_BarE(A) and η:C_BarE(A)→ C_min(A) a pair of comparison morphisms between both resolutions:


[  … A⊗_Ek𝒜_2⊗_EA            d_2⟶      A⊗_Ekℛ⊗_EA            d_1⟶    A⊗_EkQ_1⊗_EA            d_0⟶           A⊗_EA            →  0                                ;      _φ_3↓↑_η_3                      _φ_2↓↑_η_2                      _φ_1↓↑_η_1                      _φ_0↓↑_η_0                                ; … A⊗_EA^⊗ 3⊗_EA            δ_2⟶  A⊗_EA^⊗ 2⊗_E A            δ_1⟶       A⊗_EA⊗_EA            δ_0⟶          A⊗_E A            → 0.                                ;                 ]




After applying the functor _A^e(-,A) and considering the natural identifications at the beginning of Section 3  we obtain 
the maps φ^∗ and η^∗ and the following diagram:



[              0 ⟶       _E^e (E,A)             D_0⟶    _E^e( kQ_1,A)             D_1⟶    _E^e( kℛ, A )             D_2⟶   _E^e( k𝒜_2,A)…                 ;                    _η_0^∗↓↑_φ_0^∗                    _η_1^∗↓↑_φ_1^∗                    _η_2^∗↓↑_φ_2^∗                    _η_3^∗↓↑_φ_3^∗                                  ;              0 ⟶       _E^e (E,A)           δ_0^∗⟶        _E^e(A,A)           δ_1^∗⟶    _E^e(A⊗_EA,A)           δ_2^∗⟶ _E^e(A^⊗_E 3,A)…                                  ;                  ]




The maps φ^* and η^* induce isomorphisms at the cohomology level that we will still denote φ^* and η^*.
We start the description of the morphisms of complexes φ:C_min(A)→ C_BarE(A) and η:C_BarE(A)→ C_min(A).
    Firstly, φ_0 and η_0 can both be chosen as the identity of A⊗_EA. We will construct φ_i for i>0 using the homotopy t_∗ defined in Lemma <ref>.

Given i∈ℕ and 1⊗_Eu⊗_E1∈ P_i-1, we define:

    φ_i(1⊗_Eu⊗_E1)=t_i-1φ_i-1d_i-1(1⊗_Eu⊗_E1).



and the we extend φ_i as an A-bimodule morphism. Given λ, μ∈ℬ:

    φ_i(λ⊗_Eu⊗_Eμ)=λ t_i-1φ_i-1d_i-1(1⊗_Eu⊗_E1)μ.

Finally we extend it linearly.



The family of functions φ={φ_i}_i∈ℕ∪{0} is a morphism of complexes.


To prove that φ is a morphism of complexes we need to verify that the corresponding diagrams commute, that means 
that δ_iφ_i+1=φ_id_i for all i. We will prove it by induction. 
The case i=0 is clear. For the inductive step, let us suppose that  δ_j-1φ_j=φ_j-1d_j-1 for 
any j> 0 and prove that δ_jφ_j+1=φ_jd_j. Firstly, observe that it is sufficient to 
prove the result for 1⊗_Eu⊗_E1∈ P_j since φ_i, d_i and δ_i are morphisms of 
A-bimodules for all i.

By definition of φ_j+1,

    δ_jφ_j+1(1⊗_Eu⊗_E1)   =δ_jt_jφ_jd_j(1⊗_Eu⊗_E1)
       =
    (Id-t_j-1δ_j-1)φ_jd_j(1⊗_Eu⊗_E1).





The inductive hypothesis says that δ_j-1φ_j=φ_j-1d_j-1, as a consequence
 
    t_j-1δ_j-1φ_jd_j(1⊗_Eu⊗_E1)=t_j-1φ_j-1d_j-1d_j(1⊗_Eu⊗_E1)

 which is zero since d_j-1d_j=0, and we obtain the desired equality.



In lower degrees, using the inductive definition, we obtain the explicit formulas:


    φ_1(λ⊗_Eα⊗_Eμ)=λ⊗_Eα⊗_Eμ.



    φ_2(λ⊗_Er⊗_Eμ)=∑λ⊗_Er^(1)⊗_Er^(2)⊗_Er^(3)μ.



    φ_3(λ⊗_E u⊗_Eμ)=∑λ⊗_Ew_2⊗_E(w_1w_0)^(1)⊗_E(w_1w_0)^(2)⊗_E(w_1w_0)^(3)μ

where u=w_2w_1w_0 as a right 2-ambiguity.



For the definition of {η_i}_i∈ℕ we will use a similar procedure, this time with
the homotopy {s_i}_i≥ -1 constructed in Proposition <ref> for our minimal resolution.
Namely, given i∈ℕ and 
1⊗_Ea_1⊗_E…⊗_Ea_i⊗_E 1∈ A⊗_EA^⊗_E i⊗_EA:

    η_i(1⊗_Ea_1⊗_E…⊗_Ea_i⊗_E 1)=s_i-1η_i-1δ_i-1(1⊗_Ea_1⊗_E…⊗_Ea_i⊗_E 1),

and we extend η_i as a morphism of A-bimodules and then linearly.




The proof of the following proposition is analogous to  the previous one and will be omitted. 


The family of functions η={η_i}_i∈ℕ is a morphism of complexes.




Let us obtain the explicit formulas of η_i for lower degrees.
For i=1, and 1⊗_Ea⊗_E1 ∈ A⊗_EA⊗_EA:

    η_1(1⊗_Ea⊗_E1)   =s_0η_0(a⊗_E1-1⊗_Ea)
       =
    s_0(a⊗_E1-1⊗_Ea)
       =∑ a^(1)⊗_Ea^(2)⊗_Ea^(3),

since s_0(a⊗_E1)=0.

We continue now with η_2. Given 1⊗_Ea_1⊗_Ea_2⊗_E1∈ A⊗_EA^⊗_E 2⊗_EA with a_1, a_2∈ℬ=ℬ-{e_x:x∈ Q_0},

    η_2(1⊗_Ea_1⊗_Ea_2⊗_E1)   =s_1η_1δ_1(1⊗_Ea_1⊗_Ea_2⊗_E1)
       =
    s_1η_1(a_1⊗_Ea_2⊗_E1-1⊗_Ea_1a_2⊗_E1+1⊗_Ea_1⊗_Ea_2).


There are three different cases to consider:


  * If a_1a_2∈ℬ, then

    η_1δ_1(1⊗_Ea_1⊗_Ea_2⊗_E1)   =∑ a_1a_2^(1)⊗_Ea_2^(2)⊗_Ea_2^(3)
       -∑ (a_1a_2)^(1)⊗_E(a_1a_2)^(2)⊗_E(a_1a_2)^(3)
       +∑ a_1^(1)⊗_Ea_1^(2)⊗_Ea_1^(3)a_2=0.


So, if a_1a_2∈ℬ: 
η_2(1⊗_Ea_1⊗_Ea_2⊗_E1)=0.


  * If a_1a_2 contains at least one monomial relation, let {σ_1, σ_2,… ,σ_p} be the set of 
monomial relations that appear in a_1a_2, ordered by their starting point in a_1. For every σ_i we write 
a_1a_2=(σ_i)^lσ_i(σ_i)^r where (σ_i)^l is the part of a_1a_2 on the left of 
σ_i and (σ_i)^r is the right part.
    By definition of  η_1,

    η_2(1⊗_Ea_1⊗_Ea_2⊗_E1)   =∑ s_1(a_1a_2^(1)⊗_Ea_2^(2)⊗_Ea_2^(3))
       +∑ s_1(a_1^(1)⊗_Ea_1^(2)⊗_Ea_1^(3)a_2).


The first summand is zero since a_2∈ℬ and this implies that no monomial relation in a_1a_2 starts 
after the end of a_1. With respect to the second summand, observe that if a_1^(2) is an arrow that starts 
before the first arrow of σ_p, then a_1^(3)a_2=0 since it contains σ_p. In conclusion, if a_1a_2 
contains a monomial relation, then:

    η_2(1⊗_Ea_1⊗_Ea_2⊗_E1)=(σ_p)^l⊗_Eσ_p⊗_E(σ_p)^r.




  * If a_1a_2=W_ρ_i for some relation ρ_i∈ R_nomon, then
η_2(1⊗_Ea_1⊗_Ea_2⊗_E1) equals
0.9 s_1(∑ a_1a_2^(1)⊗_Ea_2^(2)⊗_Ea_2^(3)-∑ (f_ρ_i)^(1)⊗_E(f_ρ_i)^(2)⊗_E(f_ρ_i)^(3)+∑ a_1^(1)⊗_Ea_1^(2)⊗_Ea_1^(3)a_2).

The map s_1 applied to the first and second summands is zero. Applying s_1 to the third summand we conclude that

    η_2(1⊗_Ea_1⊗_Ea_2⊗_E1)=1⊗_Eρ_i⊗_E1.



We summarize the information obtained as follows:


    η_2(λ⊗_Ea_1⊗_Ea_2⊗_Eμ)={[                         0              if a_1a_2∈ℬ,; λ(σ_p)^l⊗_Eσ_p⊗_E(σ_p)^rμ             if a_1a_2∈ I,;               λ⊗_Eρ_i⊗_Eμ          if a_1a_2=W_ρ_i. ].




We will need the explicit formula of η_3, for the purpose of computing it we are going to consider four different cases.

Given a_1, a_2, a_3∈ℬ, let us consider 
λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ∈ A⊗_EA^⊗_E 3⊗_EA; 
the expression
η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ) is equal to: 

    λ(s_2η_2(a_1⊗_Ea_2⊗_Ea_3⊗_E1-
    1⊗_Ea_1a_2⊗_E a_3⊗_E1+1⊗_Ea_1⊗_Ea_2a_3⊗_E1-1⊗_Ea_1⊗_Ea_2⊗_Ea_3))μ.




  * We will start with the case where a_1a_2∈ℬ and a_2a_3∈ℬ.
    Applying the definition of η_2 in the first and last summands, they vanish, so we obtain that 
    
    η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ)=λ(s_2(-η_2(1⊗_Ea_1a_2⊗_E a_3⊗_E1)+η_2(1⊗_Ea_1⊗_Ea_2a_3⊗_E1)))μ.

    Observe that in this case 
    η_2(1⊗_Ea_1a_2⊗_E a_3⊗_E1)=η_2(1⊗_Ea_1⊗_Ea_2a_3⊗_E1) 
    since, by definition of η_2 both terms depend on a_1a_2a_3.















In conclusion, if a_1a_2∈ℬ and a_2a_3∈ℬ, then:

    η_3(λ⊗_Ea_1⊗_Ea_2⊗_E a_3⊗_Eμ)=0.



  * Let us now consider the case where a_1a_2∈ℬ and a_2a_3∈ I. In this case,
 
    η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ)=λ(s_2η_2(a_1⊗_Ea_2⊗_Ea_3⊗_E1-
    1⊗_Ea_1a_2⊗_E a_3⊗_E1))μ.

By definition of η_2 we obtain that
    
    η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ)=λ(s_2(a_1(σ_p)^l⊗_Eσ_p⊗_E
    (σ_p)^r-
        a_1(σ_p)^l⊗_Eσ_p⊗_E(σ_p)^r))μ

    since the last relation of a_2a_3 starting from the left coincides with the last one of a_1a_2a_3.
    In conclusion, if a_1a_2∈ℬ and a_2a_3∈ I, then:
    
    η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ)=0.



  * Let us continue with the symmetric case, this means a_1a_2∈ I and a_2a_3∈ℬ.

    η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ)=λ(s_2η_2(1⊗_Ea_1⊗_Ea_2a_3⊗_E1-
    1⊗_Ea_1⊗_Ea_2⊗_Ea_3))μ.

Using the definition of η_2 in this case we obtain

    η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ)=λ(s_2((δ_r)^l⊗_Eδ_r⊗_E(δ_r)^r-
    (σ_q)^l⊗_Eσ_q⊗_E(σ_q)^ra_3))μ

where δ_r is the last relation in a_1a_2a_3 and σ_q is the last relation of a_1a_2 both starting from 
the left.

Observe that the first term vanishes since δ_r(δ_r)^r does not contain 2-ambiguities.
Applying s_2 to the second term we get
        
    λ(∑ (σ_q)^l(σ_qσ_q^ra_3)^(1)⊗_E(σ_qσ_q^ra_3)^(2)⊗_E(σ_qσ_q^ra_3)^(3))μ

 with (σ_qσ_q^ra_3)^(2)∈𝒜_2. Notice that the sum is zero if there is no 2-ambiguity contained 
 in σ_qσ_q^ra_3.
 
 Consider the set {σ_1,… ,σ_p} of the monomial relations contained in a_1a_2a_3 ordered by their starting point in a_1. For every 1≤ i≤ p-1, we will call w_i the 2-ambiguity obtained from σ_i and σ_i+1.
Rewriting (<ref>), if a_1a_2∈ I and a_2a_3∈ℬ then

    η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ)=∑_i=q^p-1λ(w_i)^l⊗_Ew_i⊗_E(w_i)^rμ.



  * Lastly we will analyse the case where a_1a_2∈ I and a_2a_3∈ I.
Here, after applying the definition of η_2,

    η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ)=λ(s_2(a_1δ_r^l⊗_Eδ_r⊗_Eδ_r^r-
    σ_q^l⊗_Eσ_q⊗_Eσ_q^ra_3))μ

where δ_r is the last relation of a_2a_3 and σ_q is the last relation of a_1a_2.

If δ_r and σ_q are disjoint, the expressions a_1δ_r^l⊗_Eδ_r⊗_Eδ_r^r and 
σ_q^l⊗_Eσ_q⊗_Eσ_q^ra_3 must be zero.
Now consider the case where δ_r and σ_q are not disjoint. The first term vanishes since δ_rδ_r^r 
does not contain 2-ambiguities.
Let {σ_1,… ,σ_p} be the set of monomial relations in a_1a_2a_3, ordered by their starting point in 
a_1a_2. We have,

    η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ)=∑_i=q^p-1λ(w_i)^l⊗_Ew_i⊗_E(w_i)^rμ

where w_i is the 2-ambiguity that contains the relations σ_i and σ_i+1.





Summarising,

η_3(λ⊗_Ea_1⊗_Ea_2⊗_Ea_3⊗_Eμ)={[ λ∑_i=q^p-1(w_i)^l⊗_Ew_i⊗_E(w_i)^rμ if  a_1a_2∈ I
      and δ_r∩σ_q≠∅,;                                  0                         otherwise, ].
where {w_i}_i=q,…, p-1 is the set of 2-ambiguities contained in a_1a_2a_3, such that 
s(w_i)≥ s(σ_q) with σ_q the last relation of a_1a_2 and δ_r the last relation of 
a_1a_2a_3.

The explicit formulas of the comparison morphisms in higher degrees are usually very hard to find. In spite of that, the next 
propositions will help us later to describe ^i(A) as Lie representation of ^1(A) for i≥ 2.


Let i∈ℕ and w=u_0… u_i-1=w_i-1… w_0 an (i-1)-ambiguity. The comparison morphism φ_i is 
such that

    φ_i(1⊗_E w⊗_E 1)=∑_a^(1)… a^(i+1)=w1⊗_Ea^(1)⊗_E…⊗_Ea^(i)⊗_Ea^(i+1)

where a^(j) with 1≤ j≤ i+1 are non zero paths in A.



For the cases i=1, 2, 3 we use the explicit formulas. Let us now suppose that the result holds for some even i-1∈ℕ. Using the formula of φ_i,

    φ_i(1⊗_Ew⊗_E1)   = t_i-1φ_i-1d_i-1(1⊗_Ew⊗_E1)
       =t_i-1φ_i-1(w_i-1⊗_Ew_i-2… w_0⊗_E1)
       -t_i-1φ_i-1(1⊗_Eu_0… u_i-2⊗_Eu_i-1).

Observe that the second summand is zero in the E-reduced Bar resolution due to the inductive hypothesis and the definition of 
t_i-1.
Finally, the inductive hypothesis for φ_i-1(1⊗_Ew_i-2… w_0⊗_E 1) and the definition of t_i-1, give the result.

When i-1 is odd,

    φ_i(1⊗_Ew⊗_E1)   =t_i-1φ_i-1d_i-1(1⊗_Ew⊗_E1)
       
    =∑_wt_i-1(w^(1)φ_i-1(1⊗_Ew^(2)⊗_E1)w^(3))
       =∑_w∑_b^(1)… b^(i)=w^(2)t_i-1(w^(1)⊗_Eb^(1)⊗_E…⊗_Eb^(i-1)⊗_Eb^(i)w^(3))
       =∑_w∑_b^(1)… b^(i)=w^(2)1⊗_Ew^(1)⊗_Eb^(1)⊗_E…⊗_Eb^(i-1)⊗_Eb^(i)w^(3),

observing that w^(1)b^(1)… b^(i)w^(3)=w, the proof is complete.





  * For any i≥ 3 and a_1,…, a_i in ℬ  

    η_i(1⊗_E a_1⊗_E…⊗_Ea_i⊗_E 1)=∑_aα_123a^(1)⊗_Ea^(2)⊗_Ea^(3)

with α_123∈k, a^(1)a^(2)a^(3)=a=a_1… a_i and a^(2)∈𝒜_i-1.

In particular, if the path a_1… a_i does not contain (i-1)-ambiguities, 

    η_i(1⊗_E a_1⊗_E…⊗_Ea_i⊗_E 1)=0.


  * For all i≥ 1 and w=w_0… w_i-1=z_i-1… z_0 an (i-1)-ambiguity: 

    η_i(1⊗_Ew_0⊗_E…⊗_Ew_i-1⊗_E 1)=1⊗_E w⊗_E 1.




(1) The case i=3 has already been checked. For the inductive step,

    η_i(1⊗_E a_1⊗_E…⊗_Ea_i⊗_E 1)   =s_i-1η_i-1δ_i-1(1⊗_E a_1⊗_E a_2⊗_E…⊗_Ea_i⊗_E 1)
       =a_1s_i-1η_i-1(1⊗_Ea_2⊗_E …⊗_Ea_i⊗_E 1)
       +∑_j=1^i-1(-1)^js_i-1η_i-1(1⊗_Ea_1⊗_E…⊗_Ea_ja_j+1⊗_E…⊗_E1)
       +(-1)^is_i-1(η_i-1(1⊗_Ea_1⊗_E…⊗_E a_i-1⊗_E1)a_i).

Using the inductive hypothesis and the definition of s_i-1 in every summand, the result is obtained.



(2) Firstly observe that since w_j, z_k∈ℬ for 0≤ j,k≤ i-1 it will not be necessary to 
take classes of w_j or z_k in A.

The case where i=1 is clear using the definition of η_1.
For i=2, the explicit formula of η_2 gives

    η_2(1⊗_Ew_0⊗_E w_1⊗_E 1)=(σ_p)^l⊗_Eσ_p⊗_E(σ_p)^r

 where σ_p is the last monomial relation from the left that appears in w_0w_1. Since, by  definition of ambiguity, 
 w_0w_1 cannot strictly contain a monomial relation, σ_p=w_0w_1 and the result is proven.

For the inductive step,

    η_i(1⊗_E w_0⊗_E…⊗_E w_i-1⊗_E 1)   =s_i-1η_i-1δ_i-1(1⊗_E w_0⊗_E…⊗_E w_i-1⊗_E 1)
       =w_0s_i-1η_i-1(1⊗_E…⊗_E w_i-1⊗_E 1)
       + ∑_j=0^i-2(-1)^j+1s_i-1η_i-1(1⊗_E…⊗_E w_jw_j+1⊗_E…⊗_E 1)
       + (-1)^is_i-1η_i-1(1⊗_E w_0⊗_E…⊗_E w_i-1).

The summands in the second line vanish since w_jw_j+1∈ I for all j=0,… ,i-2.
Using the inductive hypothesis for the last summand it turns out that it is equal to

    (-1)^is_i-1(1⊗_E w_0w_1… w_i-2⊗_E w_i-1)

and using the definition of s_i-1 the expression equals to 1⊗_E w ⊗_E 1, since w is the only (i-1)-ambiguity 
contained in w_0… w_i-1.

The last step is to prove that the first summand vanishes. Using the first part of this proposition, the term 
η_i-1(1⊗_E w_1⊗_E…⊗_E w_i-1⊗_E 1) will be a linear combination of elements of the 
form a^(1)⊗_E a^(2)⊗_E a^(3) with a=a^(1) a^(2)a^(3) a path from s(w_1) to t(w_i-1) 
and a^(2) an (i-2)-ambiguity. Since the path w_1 does not start at 0 and the quiver is toupie, the only path 
from s(w_1) to t(w_i-1) is w_1… w_i-1. The result of applying s_i-1 to the linear combination vanishes 
since a^(1) a^(2)a^(3)=w_1… w_i-1 and it cannot contain an (i-1)-ambiguity.











For any toupie algebra A, η is a left inverse of φ.


Again, we will prove the result by induction.
The cases i=0 and i=1 are almost immediate using the corresponding explicit formulas.
Consider i≥ 2. Observe that it is sufficient to verify the equality for the elements of the form 
1⊗_Eu⊗_E1 ∈ P_i.

Let us check the result for i=2. Given a monomial relation σ_i,

    η_2φ_2(1⊗_Eσ_i⊗_E 1)   =η_2(∑_σ_i1⊗_Eσ_i^(1)⊗_Eσ_i^(2)⊗_Eσ_i^(3))
       =
    1⊗_Eσ_i⊗_E 1

since, by definition of η_2, the only summand which is not zero is when σ_i^(3)=1.

Let us now consider a non monomial relation of the form
 
    ρ_i=α^(k_i)+∑_j>k_ib_ijα^(j).

In this case,

    η_2φ_2(1⊗_Eρ_i⊗_E 1)   =∑_α^(k_i)η_2(1⊗_E(α^(k_i))^(1)⊗_E(α^(k_i))^(2)⊗_E
    (α^(k_i))^(3))
       +∑_j>k_ib_ij∑_α^(j)η_2(1⊗_E(α^(j))^(1)⊗_E(α^(j))^(2)⊗_E
    (α^(j))^(3)).

In the first sum, the only summand which is not zero is when
(α^(k_i))^(3)=1 and in this case the result is 1⊗_Eρ_i⊗_E 1. 
The second sum is zero by definition of η_2, so we obtain the equality.


Now consider the case i≥ 3. If w=w_0… w_i-1=u_i-1… u_0 is an (i-1)-ambiguity, then

    η_iφ_i(1⊗_Ew⊗_E1)   =s_i-1η_i-1δ_i-1t_i-1φ_i-1d_i-1(1⊗_Ew
    ⊗_E1).

The map t_* is a contracting homotopy for the E-reduced Bar complex, 
    δ_i-1t_i-1=Id-t_i-2δ_i-2
 and 
the right hand side of the previous equality is

    s_i-1η_i-1φ_i-1d_i-1(1⊗_Ew⊗_E1)-s_i-1η_i-1t_i-2δ_i-2φ_i-1d_i-1(1⊗_Ew⊗_E1).

Since φ is a morphism of complexes from C_min(A) to C_BarE(A) we know that 
δ_i-2φ_i-1d_i-1=φ_i-2d_i-2d_i-1=0 and this implies that the second term is zero. 
Using the inductive hypothesis in the first term it is sufficient to prove that

    s_i-1d_i-1(1⊗_Ew⊗_E1)=1⊗_Ew⊗_E1.

For i odd,

    s_i-1d_i-1(1⊗_Ew⊗_E1)   =u_i-1s_i-1(1⊗_Eu_i-2… u_0⊗_E1)
       -s_i-1(1⊗_Ew_0… w_i-2⊗_Ew_i-1)
       =1⊗_Ew⊗_E1.

For i even,

    s_i-1d_i-1(1⊗_Ew⊗_E1)   =s_i-1(∑_w^(2)∈ V^(i-2)w^(1)⊗_Ew^(2)⊗_Ew^(3))
       =1⊗_Ew⊗_E1

and the result is proven.


















§ GERSTENHABER STRUCTURE OF ^*(A)


In this section we fix k=ℂ. We describe the Lie structure of ^1(A), while in the next sections we will
provide the structure of ^i(A), with i≥ 2, as Lie representations of ^1(A).


 §.§ Cup product

The cup product of the Hochschild cohomology of a toupie algebra is trivial, see <cit.>. For the convenience of the reader we sketch an alternative proof.

Given a toupie algebra A, f∈^n(A) and g∈^m(A) with m,n>0, the cup product f⌣g vanishes in ^m+n(A).



We will prove that f⌣_redg =0 in ^m+n(A). 
Since η^* is an isomorphism at the cohomology level, there exist F∈_E^e(k𝒜_n-1,A) and 
G∈_E^e(k𝒜_m-1,A) such that 

    f=η_n^*F=η_n^*F,


    g=η_m^*G=η_m^*G.

We choose f' and g' such that f'=η_n^*F and g'=η_m^*G

Let us suppose that f'⌣_red g'≠ 0, that is f'⌣_red g' is not a coboundary  and in particular there 
exists a_1⊗…⊗ a_n+m∈A^⊗_E(m+n) such that

    f'⌣_red g'(a_1⊗…⊗ a_n+m)=f'(a_1⊗…⊗ a_n)g'(a_n+1⊗…⊗ a_n+m)≠ 0.

However, 
    f'(a_1⊗…⊗ a_n)=η_n^*(F)(a_1⊗…⊗ a_n)=F(η_n(1⊗ a_1⊗…⊗ a_n⊗ 1))
 
and 

    g'(a_n+1⊗…⊗ a_n+m)=η_m^*(G)(a_n+1⊗…⊗ a_n+m)=G(η_m(1⊗ a_n+1⊗…⊗ a_n+m⊗ 1)),
 
so f'(a_1⊗…⊗ a_n) and g'(a_n+1⊗…⊗ a_n+m) must be linear combinations of paths of 
positive length that start at 0, so their cup product will be zero, which leads us to a contradiction. 
In conclusion, f⌣_redg=f'⌣_red g'=0.




 §.§ Gerstenhaber bracket in higher degrees

Next we will compute the Gerstenhaber bracket of two elements in the higher cohomology spaces of A. We will prove that the bracket 
of two cocycles of degree greater than one is always zero due to the particular shape of the quiver.

Let A=kQ/I be a toupie algebra, f∈^n(A) and g∈^m(A) with m,n>1.
The Gerstenhaber bracket [f, g]_red vanishes in ^m+n-1(A).


We want to prove that [f,g]_red=0 in ^m+n-1(A). As before we choose f' and g' such 
that f'=η_n^*F and g'=η_m^*G
with F and G obtained from the minimal resolution.
Let us suppose that there exists a_1⊗…⊗ a_n+m-1∈A^⊗_E(m+n-1) such that 
[f',g']_red(a_1⊗…⊗ a_n+m-1)≠ 0.
Recall that
[f',g']_red=f'∘ g'-(-1)^(m-1)(n-1)g'∘ f'
and that f'∘ g'(a_1⊗…⊗ a_n+m-1) is

    ∑_i=1^n(-1)^(i-1)(m-1)f'(a_1⊗⋯ a_i-1⊗g'(a_i⊗…⊗ a_i+m-1)⊗ a_i+m⊗…⊗ a_n+m-1).

We will verify that each summand is zero. Suppose that g'(a_i⊗…⊗ a_i+m-1)≠ 0 in A. 
Since g'(a_i⊗…⊗ a_i+m-1)=G( η_m(1⊗ a_i⊗…⊗ a_i+m-1⊗ 1)) we know that it 
will be a linear combination of paths from 0 to ω that is 
g'(a_i⊗…⊗ a_i+m-1)=∑_α^(j)∈_0ℬ_ωλ_jα^(j). 
In this case 
a_1⊗_E…⊗_Ea_i-1⊗_Eα^(j)⊗_Ea_i+m⊗_E…⊗_Ea_n+m-1 
vanishes in A^⊗_En, so f'∘ g'=0. 
A similar argument proves that g'∘ f'=0 and we conclude that
[f,g]_red=[f',g']_red=0 in ^m+n-1(A).




 §.§ Gerstenhaber bracket in ^1(A)

From now on we will identify the elements of ^i(A) with their classes when there is no confusion.
We will use the notation of <cit.> that we recall.
Given α h∈_E^e(kQ_1, A) and a path ã in A, we will denote ã^α h the sum 
of all the non zero paths obtained replacing every appearance of α in ã by h. If the path ã does not 
contain the arrow α or if when we replace α in ã by h we get 0∈ A, we define 
ã^α h= 0. Observe that for toupie algebras any element of type ã^α h will have at most 
one summand.


Given α h∈_E^e(kQ_1, A), b∈ℬ and f∈_E^e(A, A), the 
following equalities hold:


  * η_1^∗(α h)(b)=b^α h.

  * φ_1^∗(f)=∑_α∈ Q_1∑_αc ∈ Q_1ℬλ_α, cαc, if 
f(α)=∑_c∈ℬλ_α, cc.




(1) Given αh ∈_E^e(kQ_1, A), we will denote by (αh) the element in 
_A^e(A⊗_EkQ_1⊗_EA, A) obtained from αh using the canonical identification of 
_E^e(kQ_1, A) with _A^e(A⊗_EkQ_1⊗_EA, A). 

For b∈ℬ,

    η_1^∗(αh)(b)=(αh)η_1(1⊗_E b⊗_E 1)=(αh)∑ b^(1)⊗_Eb^(2)⊗_E b^(3)=∑ b^(1)(αh)(b^(2))b^(3).



  * If α is an arrow contained in b, that is b=α^lαα^r, then η_1^∗(αh)(b)=α^lh α^r.

  * If not, then η_1^∗(αh)(b)=0. 

The first equality is proven.




(2) Given f∈_E^e(A, A) and α∈ Q_1, we will denote by f̃ the element in 
_A^e(A⊗_EA⊗_EA, A) associated to f using the canonical identification of 
_E^e(A, A) with _A^e(A⊗_EA⊗_EA, A). Now, 

    φ_1^*(f)(α)=f̃(φ_1(1⊗_Eα⊗_E1))
    =f̃(1⊗_Eα⊗_E1)=f(α).

If f(α)=∑_c∈ℬλ_α, cc, then
we conclude that φ_1^∗(f)=∑_α∈ Q_1∑_αc ∈ Q_1ℬλ_α, cαc.

The next theorem gives the formula of the Gerstenhaber bracket restricted to ^1(A). The corresponding result for monomial 
algebras has been proven in <cit.>. Even if the elements of ^1(A) here differ from those studied by Strametz, the formula 
for the bracket can be written in a similar way.

Let A be a toupie algebra.
The Gerstenhaber bracket in ^1(A) can be expressed in terms of the minimal resolution as:

    [αh, β b]=β b^αh-αh^β b,

with αh, βb∈_E^e(kQ_1, A).


Given αh, βb∈_E^e(kQ_1, A), let us compute 
[αh, β b]∈_E^e(kQ_1, A) using the comparison morphisms.
Given γ∈ Q_1,

    [αh, β b](γ)   =φ^∗_1[η_1^∗(αh), η_1^∗(βb)](γ)
       =[η_1^∗(αh), η_1^∗(βb)]φ_1(1⊗_Eγ⊗_E1)
       =[η_1^∗(αh), η_1^∗(βb)](γ).


Applying now the definition of the Gerstenhaber bracket, we get:


    [η_1^∗(αh), η_1^∗(βb)](γ)   =η_1^∗(αh)∘η_1^∗(βb)(γ)-η_1^∗(βb)∘η_1^∗(αh)(γ)
       =(γ^βb)^αh-(γ^αh)^βb.

There are three cases to consider.


  * If γ=β, then γ^βb=b, and if also b contains α, then we replace α by h in b.

  * If γ=α, then γ^αh=h, and if also h contains β, then we replace β by b in h.

  * If γ≠α and γ≠β, then [αh, β b](γ)=0.



In conclusion, [αh, β b]=β b^αh-αh^β b.

The computation we have just made in terms of the complex induces the formula of the Gerstenhaber bracket in ^1(A).


If αα and ββ belong to ^1(A), then [αα, ββ]=0.




§ DECOMPOSITION OF ^1(A) AS A LIE ALGEBRA



In this section we will give a description of ^1(A) as a Lie algebra. We will first find necessary and sufficient 
conditions for A to be abelian and next we will describe in detail the centre of ^1(A).
 
We will use the following notation for the explicit basis of ^1(A) computed in Subsection <ref>:



  * y_i=α^i_0α^i_0   with    α^(i)  a branch containing monomial relations.

  * w_pq=α^(p)α^(q) for p≠ q and p,q=1… ,a.

  * z_us=α^(u)α^(s) for u=1,… ,a and α^(s)∈_0ℬ_ω-Z.

  * x_j=α^(j)α^(j)-α^(1)α^(1) for j=2… ,a.

  * t_k=∑_α^(i)∈ Q_ρ^kα^i_0α^i_0 for k=1,… ,r.


Recall that C”_1={y_i:α^(i) is a branch with monomial relations}.


Using Theorem <ref> we compute the Gerstenhaber brackets of the elements of the basis of ^1(A) and obtain the 
following table:



where

 
  * A_j^p'q'=[x_j, w_p'q']=δ_j,q'w_p'q'-δ_j,p'w_p'q'-δ_q',1w_p'1+δ_1,p'w_1q',

  * B_j^u's'=[x_j,z_u's']=-δ_j,u'z_u's'+δ_u',1z_1s',

  * C_k^u's'=[t_k,z_u's']= {[        z_u's' α^(s')∈ Q_ρ^k;            0,               ].


  * D_pq^u's'=[w_pq,z_u's']=-δ_q,u'z_ps'
and


  * E_pq^p'q'=[w_pq, w_p'q']={[               δ_p,q'w_p'q-δ_q,p'w_pq'             if  q≠ p'
      or  p≠ q'; δ_p,q'α^(p')α^(p')-δ_q,p'α^(q')α^(q')                  if  q=p'  and  p=q'. ].

In the last case, where q=p' and p=q', we can distinguish three different cases:

E_pp'^p'p
={[       x_p'    if  p=1;       -x_p   if  p'=1;   x_p-x_p' otherwise. ].






Let A be a toupie algebra. The Lie algebra ^1(A) is abelian if and only if a=0 or D≤ 1. Moreover, if a=0, then the 
enveloping algebra 𝒰(^1(A)), is isomorphic to the polynomial algebra k[x_1,⋯ ,x_r+m-1]. 
In case a=D=1, we have that 𝒰(^1(A)) is isomorphic to k[y_1,⋯ ,y_m].


For the first statement, recall from Section <ref> that:

    _E^e(kQ_1,A)=k(Q_1-Z Q_1-Z)+ k(Z_0ℬ_ω).

If a=0, then Z is empty and _E^e(kQ_1,A) is generated by the elements of the form αα for 
some α∈ Q_1. Using Remark <ref>, we conclude that ^1(A) is abelian.

If D≤ 1, we distinguish two different cases:


  * if a=0, we argue as before,

  * if a=1, then Z={α^(1)} and _E^e(kQ_1,A) is generated by the elements of the form 
αα with α∈ Q_1-Z together with {α^(1)α^(1)}. Using again Remark <ref> we conclude 
that ^1(A) is abelian.

For the converse, suppose that a>0 and D>1.
Again, we consider two different cases:


  * If D=a, then there are at least two elements in Z that we will call α^(1) and α^(2) and the Lie bracket table gives:

    [w_21, w_12]=-x_2≠ 0.


  * If D> a>0, then there exists at least one element in Z that we will call α^(1) and one element in 
_0ℬ_ω-Z that we will call α^(p) which belongs to the connected component Q_ρ^k for some k. 
Consider the elements 

z_1p and t_k. Using again the computations in the Lie bracket table:

    [z_1p, t_k]=-z_1p≠ 0.


If a=0, Remark <ref> implies that dim_k^1(A)=r+m-1 and since ^1(A) is abelian, 
the enveloping algebra 𝒰(^1(A)) is isomorphic to k[x_1,⋯ ,x_r+m-1]. If a=D=1, then r=0 and 
Da=1 so again by Remark <ref>,  dim_k^1(A)=m and 
𝒰(^1(A))≃k[y_1,⋯ ,y_m].


From now on we will suppose that a>0 and D>1.

We will next describe the centre of the Lie algebra ^1(A).

Let A be a toupie algebra
with a>0. 
The centre of the Lie algebra ^1(A) is the k-vector space spanned by C”_1={y_i:i=1,… ,m}.



Using the Lie bracket table we check that C”_1⊂𝒵(^1(A)).
Next  we will prove that every element of the centre of the algebra can be written as a linear combination of elements 
of C”_1.

Let Γ be an element in 𝒵(^1(A)). Write Γ in terms of the given basis of ^1(A) as follows:

    Γ=∑_jA_jx_j+∑_p,qB_p,qw_pq+∑_u,sC_u,sz_us+∑_kD_kt_k+∑_iE_iy_i.

Since the bracket of Γ with any element of C”_1 is 0, we determine the restrictions 
that this condition imposes on the coefficients of Γ.
For every k,

    0=[Γ, t_k]=[∑_u,s C_u,sz_us, t_k]=∑_u,s C_u,s[z_us, t_k]=-∑_u,s C_u,sz_us.

The z_us's are linearly independent, so C_u,s vanishes for every s and for every u=1,…,a.We also know that for every j=2,…,a:

    0=[Γ, x_j]=[Γ, α^(j)α^(j)]-[Γ, α^(1)α^(1)]=
    [∑ B_p,qw_pq, α^(j)α^(j)]-[∑ B_p,qw_pq, α^(1)α^(1)],
 
using Remark <ref> and the fact that the coefficients C_r,s vanish.
Now:

    [Γ, x_j]   =-∑_p(B_p,jw_pj-B_p,1w_p1)+∑_q(B_j,qw_jq-B_1,qw_1q)
       =
    -2B_1,jw_1j+2B_j,1w_j1-∑_p≠ 1B_p,jw_pj
           +∑_q≠ 1B_j,qw_jq+∑_p≠ jB_p,1w_p1-∑_q≠ jB_1,qw_1q.

Since the w_pq's are linearly independent, we conclude that B_p,q=0 for every p≠ q and p,q=1,…, a. 


We will now see that A_j=0 for j=2,…,a.
Choosing w_pq with p≠ q and 1≤ p,q≤ a, it turns out that

    0=[Γ, w_pq]=∑_j A_j[x_j,w_pq].

If p,q≠ 1 then,

    0=[Γ, w_pq]=(A_q-A_p)w_pq

so we conclude that A_p=A_q. If p=1, then:

    0   =[Γ, w_1q]=∑_j A_j[α^(j)α^(j),w_1q]-∑_j A_j[α^(1)α^(1),w_1q]
       =A_qw_1q+∑_j A_jw_1q=aA_qw_1q,

which implies A_q=0 for q=2,…, a.


Finally, given u and s such that 1≤ u≤ a and α^(s)∈_0ℬ_ω-Z we have:

    0=[Γ, z_us]=∑_k D_k[t_k,z_us]=∑_k D_k[t_k, α^(u)α^(s)]=D_lz_us

if α^(s) belongs to Q_ρ^l.
As a consequence, D_k=0 for every k and we conclude that 

    Γ=∑_i E_iy_i.
 

Consider the a-Kronecker quiver Q_a: it has only two vertices, one source and one sink, with a arrows from the 
source to the sink. In the next proposition we will prove that if the toupie quiver Q_A associated to the toupie algebra A has 
Q_a as a subquiver, then the Lie algebra ^1(A) contains a subalgebra isomorphic to ^1( kQ_a).


The Lie subalgebra of ^1(A) generated by

    {x_j, w_pq  with  p≠ q;  p,q=1,…, a  and
      j=2,…, a}

is isomorphic to ^1( kQ_a).



Let Q_A be the quiver of the toupie algebra A.
First note that kQ_a is a toupie algebra. Consequently, by Theorem <ref>, the space ^1( kQ_a) 
has a basis:

    {X_j, W_pq  with  p≠ q;  p,q=1,…, a  and  j=2,…, a}.

There is a morphism of k-Lie algebras from ^1( kQ_a) to ^1(A) that sends X_j to 
x_j and W_pq to w_pq for all j,p,q. This map is injective and induces an isomorphism between 
^1( kQ_a) and its image.




The proof of the next proposition is in <cit.>. We will just give the explicit isomorphism since we are going to 
use it later.

From now on, suppose k=.


The Lie subalgebra of ^1(A) generated by the set {x_j, w_pq  with  p≠ q;  p,q=1,…, a  and  j=2,…, a} is 
isomorphic to sl_a(ℂ).


The isomorphism sends w_pq to the elementary a× a matrix E_qp and x_j to the a× a diagonal matrix 
E_jj-E_11.

Now we will give a decomposition of ^1(A) as a Lie algebra.


Let Q be a toupie quiver, Q_ρ the quiver described in Definition <ref> and {Q_ρ^h :h=1,…, r} the set of 
the connected components of Q_ρ.
  The k-vector space 
  
    L=⟨ t_h, z_us: h=1,…, r;    u=1,…, a   and   s  such that α ^(s)∈_0ℬ_ω-Z ⟩
 
  is a solvable Lie ideal of ^1(A). Moreover,



  * S_1=⟨ t_h: h=1,…, r⟩ is an abelian Lie subalgebra; L_2=⟨ z_us: u=1,…, a   and   s
such that α^(s)∈_0ℬ_ω-Z⟩ is an abelian ideal, and:

    L=S_1⋉ L_2.


  * There is an isomorphism of Lie algebras ^1(A)≅⟨ C”_1⟩⊕ sl_a(ℂ)⋉ (S_1⋉ L_2).





Using the Lie bracket table we know that [t_h, t_h']=0,
[t_k, z_us ]= z_us or 0, and [z_us, z_u's']=0, so we conclude that L is a Lie subalgebra of ^1(A). 
Moreover, since [x_j, t_k]=0 and [w_pq, t_k]=0, L is also an ideal.

From the previous computations we deduce that L is solvable since [L,L]⊂ L_2 and L_2 is abelian.

We also deduce that S_1 is an abelian subalgebra and L_2 is an ideal. Besides, since L_2 is abelian, it is solvable.
Finally, since the projection π':L→ L/L_2 induces an isomorphism between S_1 and L/L_2 it turns out that

    L=S_1⋉ L_2.

On the other hand, consider the Lie algebra ^1(A):=^1(A)/𝒵(^1(A))

and the projection π:^1(A)→^1(A)/L. 
Propositions <ref> and <ref> provide an isomorphism between sl_a(ℂ) and ^1(A)/L, 
which implies that:

    ^1(A)≃ sl_a(ℂ)⋉ L

Using this isomorphism and Lemma <ref>, 

we obtain the decomposition of ^1(A).



Let A be a toupie algebra. The radical of the Lie algebra ^1(A) decomposes as follows:

    rad(^1(A))=L⊕⟨ C”_1⟩.



Since L is a solvable ideal and ⟨ C”_1⟩ is abelian, it follows that L⊕⟨ C”_1⟩ is a solvable ideal. 
Besides, the quotient of ^1(A) by L⊕⟨ C”_1⟩ is isomorphic  to sl_a(ℂ), which is semisimple.

The next corollary gives a similar decomposition to the one given in Theorem <ref>. It can be easily proven using the 
previous corollary and the Levi decomposition.


Let A be a toupie algebra. The algebra ^1(A) decomposes as follows:

    ^1(A)=(L⊕⟨ C”_1⟩)⋊ sl_a(ℂ).





Let A be a toupie algebra such that ^1(A) is not abelian. The Lie algebra ^1(A) is semisimple if and only if a=D 
and m=0.




If ^1(A) is semisimple, then rad(^1(A))=0 and # C”_1=m=0. Using Corollary <ref>, 
we know that ^1(A) is isomorphic to sl_a(ℂ), via the identification of sl_a(ℂ) with 
⟨ x_j, w_pq  :p≠ q;  p,q=1,…, a  and  j=2,…, a⟩ as in Corollary <ref>.
Since ^1(A) is not abelian, Theorem <ref> implies that there exists at least one arrow from 0 to 
ω that we will call α^(1)∈ Z.
If a was strictly smaller than D, then it would exist a branch α^(i)∈_0ℬ_ω-Z. 
Consequently, the element z_1i=α^(1)α^(i) would be non zero in ^1(A) and that leads to a contradiction.

Let us prove the converse. We know that if a=D and m=0, then the basis of ^1(A) does not contain elements of the form 
y_i, z_us or t_h and it follows that ^1(A)≃ sl_a(ℂ).





§ ^I(A) AS REPRESENTATION OF ^1(A)




 §.§ ^2(A) as a Lie representation of ^1(A)



In this section we will describe the action of the Lie algebra ^1(A) on ^2(A).
Given a path c, we will call _1c the first arrow of c.
Consider α b∈^1(A), ρ c∈^2(A) and a monomial relation σ belonging to a minimal set of generators of I. Let us compute [α b, ρ c]∈^2(A):

    [α b, ρ c](σ)   =φ_2^∗[η_1^∗(α b), η_2^∗(ρ c)](σ)
       =∑_σ[η_1^∗(α b), η_2^∗(ρ c)](1⊗σ^(1)⊗σ^(2)⊗σ^(3))
    
       =∑_σ(η_1^∗(α b)∘η_2^∗(ρ c)(σ^(1)⊗σ^(2)) )σ^(3)
           -∑_ση_2^∗(ρ c)(η_1^∗(α b)( σ^(1))⊗σ^(2))σ^(3)
           -∑_ση_2^∗(ρ c)(σ^(1)⊗η_1^∗(α b)(σ^(2)))σ^(3),

where the sums range over all possible decompositions of σ with σ^(2) an arrow.
Let us first compute the first summand:


    ∑_σ(η_1^∗(α b)∘η_2^∗(ρ c)( σ^(1)⊗σ^(2))) σ^(3)   =∑_ση_1^∗(α b)( (ρ c)(η_2(1⊗σ^(1)⊗σ^(2)⊗ 1)))σ^(3).


Since the monomial relation σ does not contain any other relation, the term η_2(1⊗σ^(1)⊗σ^(2)⊗ 1) vanishes except for the case where σ^(3) is 1,
thus

    ∑_ση_1^∗(α b)( (ρ c)(η_2(1⊗σ^(1)⊗σ^(2)⊗ 1)))σ^(3)   =η_1^∗(α b) (ρ c)(η_2(1⊗σ^(1)⊗σ^(2)⊗ 1))
       =η_1^∗(α b) (ρ c)(σ)
       =δ_ρ, ση_1^∗(α b) (c)
       =δ_ρ, σα b(η_1 (1⊗c⊗ 1))
       =δ_ρ, σ∑_cα b(c^(1)⊗c^(2)⊗ c^(3))
     
       ={[ δ_ρ, σc^αb  if α=_1c;;          0 otherwise. ].

As for the second summand,

-∑_ση_2^∗(ρ c)(η_1^∗(α b)(σ^(1))⊗σ^(2))σ^(3)={[ -∑_σ(ρ c)η_2(1⊗σ^(1)⊗σ^(2)⊗ 1)σ^(3)                           if α=_1σ;;                                   0                          otherwise, ].


since if α≠_1σ then η_1^∗(α b)(σ^(1))=(α b)(η_1(1⊗σ^(1)⊗ 1))=0.

For the computation in the first line observe that if α= _1σ, then b = α.

Taking into account that σ does not contain any other relation, we know that η_2(1⊗σ^(1)⊗σ^(2)⊗ 1) vanishes except for the case where σ^(3)=1; in that case,

    -∑_σ(ρ c)η_2(1⊗σ^(1)⊗σ^(2)⊗ 1)σ^(3)=-(ρ c)η_2(1⊗σ^(1)⊗σ^(2)⊗ 1)=-(ρ c)(σ)=-δ_σ,ρc.



Finally, we will prove that the last summand vanishes. For this, notice that 

-∑_ση_2^∗(ρ c)(1⊗σ^(1)⊗η_1^∗(α b)(σ^(2))⊗ 1)σ^(3)={[ -η_2^∗(ρ c)(1⊗1⊗σ^(1)⊗ 1)σ^(2)                      if α=_1σ;;                              0                     otherwise. ].

and observe that 1⊗1⊗σ^(1)⊗ 1 is zero in the E-reduced Bar resolution.



Given a non monomial relation,  which is necessarily  of the form 

    ρ_i=α^(k_i)+∑_j> k_i b_ijα^(j) for some i

 and using the same notations,

    [α b, ρ c](ρ_i)   =φ_2^∗[η_1^∗(α b), η_2^∗(ρ c)](ρ_i)
    
       =∑_α^(k_i)[η_1^∗(α b), η_2^∗(ρ c)](1⊗ (α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ (α^(k_i))^(3))
           +∑_j> k_i b_ij∑_α^(j)[η_1^∗(α b), η_2^∗(ρ c)](1⊗(α^(j))^(1)⊗(α^(j))^(2)⊗ (α^(j))^(3))
    
       =∑_α^(k_i)η_1^∗(α b)∘η_2^∗(ρ c)(1⊗ (α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ (α^(k_i))^(3))
           +∑_j> k_ib_ij∑_α^(j)η_1^∗(α b)∘η_2^∗(ρ c)(1⊗(α^(j))^(1)⊗(α^(j))^(2)⊗ (α^(j))^(3))
            -∑_α^(k_i)η_2^∗(ρ c)(η_1^∗(α b)((α^(k_i))^(1))⊗(α^(k_i))^(2))(α^(k_i))^(3)
             -∑_j> k_ib_ij∑_α^(j)η_2^∗(ρ c)(η_1^∗(α b)((α^(j))^(1))⊗(α^(j))^(2))(α^(j))^(3)
           -∑_α^(k_i)η_2^∗(ρ c)((α^(k_i))^(1)⊗η_1^∗(α b)((α^(k_i))^(2)))(α^(k_i))^(3)
           -∑_j> k_ib_ij∑_α^(j)η_2^∗(ρ c)((α^(j))^(1)⊗η_1^∗(α b)((α^(j))^(2)))(α^(j))^(3).


Let us compute the first summand:

    ∑_α^(k_i)η_1^∗(α b)∘η_2^∗(ρ c)(1⊗ (α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ (α^(k_i))^(3))
    =   ∑_α^(k_i)η_1^∗(α b)( (ρ c)(η_2(1⊗(α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ 1))(α^(k_i))^(3)).


The term η_2(1⊗(α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ 1) vanishes except when (α^(k_i))^(3)=1, and in this case, by definition of η_2, the result equals ρ_i.

Hence,

    ∑_α^(k_i)η_1^∗(α b)( (ρ c)(η_2(1⊗(α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ 1))(α^(k_i))^(3))   =η_1^∗(α b) (ρ c)(ρ_i)
       =δ_ρ, ρ_iη_1^∗(α b) (c)
       =δ_ρ, ρ_i(α b)(η_1 (1⊗c⊗ 1))
       ={[ δ_ρ, ρ_ic^α b        α=_1c;;             0    otherwise. ].
    .


Let us verify that the second summand is zero:

    ∑_j> k_ib_ij∑_α^(j)η_1^∗(α b)∘η_2^∗(ρ c)(1⊗(α^(j))^(1)⊗(α^(j))^(2)⊗ (α^(j))^(3))
    
       =∑_j> k_ib_ij∑_α^(j)η_1^∗(α b)( (ρ c)(η_2(1⊗(α^(j))^(1)⊗(α^(j))^(2)⊗ 1))(α^(j))^(3))
       =0

since η_2(1⊗(α^(j))^(1)⊗(α^(j))^(2)⊗ 1)=0 for every j>k_i.




To deal with the third summand, we need to distinguish two different cases.



  * If α is not the first arrow of α^(k_i) then:

    -∑_α^(k_i)η_2^∗(ρ c)(η_1^∗(α b)((α^(k_i))^(1))⊗(α^(k_i))^(2))(α^(k_i))^(3)=0.


  * If α is the first arrow of α^(k_i), which implies that b=α:

    -∑_α^(k_i)η_2^∗(ρ c)(η_1^∗(αα)((α^(k_i))^(1))⊗(α^(k_i))^(2))(α^(k_i))^(3)
       =-∑_α^(k_i)η_2^∗(ρ c)((αα)(η_1(1⊗(α^(k_i))^(1)⊗ 1)⊗(α^(k_i))^(2))(α^(k_i))^(3)
       =-∑_α^(k_i)η_2^∗(ρ c)((α^(k_i))^(1)⊗(α^(k_i))^(2))(α^(k_i))^(3)
       =-∑_α^(k_i)(ρ c)η_2(1⊗(α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ 1)(α^(k_i))^(3).

  The term η_2(1⊗(α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ 1) vanishes except for (α^(k_i))^(3)=1, and in this 
  case η_2(1⊗(α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ 1)=ρ_i .
Therefore,

    -∑_α^(k_i)(ρ c)η_2(1⊗(α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ 1)(α^(k_i))^(3)   =-(ρ c)η_2(1⊗(α^(k_i))^(1)⊗(α^(k_i))^(2)⊗ 1)
       =-(ρ c)(ρ_i)
       =-δ_ρ,ρ_ic.


An analogous computation shows that the fourth summand is zero. The situation is even easier, since in this case
 η_2(1⊗(α^(j))^(1)⊗(α^(j))^(2)⊗ 1)=0 for j>k. 
 


Finally we verify that the last two summands vanish. For the fifth summand observe that 

-∑_α^(k_i)η_2^∗(ρ c)((α^(k_i))^(1)⊗η_1^∗(α b)((α^(k_i))^(2)))(α^(k_i))^(3) equals -(ρ c)(η_2(1⊗1⊗(α^(k_i))^(2)⊗ 1 ))(α^(k_i))^(3)  if α is the first arrow of α^(k_i) and it is zero otherwise.

From the fact that 1⊗1⊗(α^(k_i))^(2)⊗ 1 is zero in the E-reduced Bar resolution,
we conclude that the fifth summand also vanishes.
The last summand also vanishes for analogous reasons.Summarising, given αb∈^1(A) and ρc∈^2(A):


    [αb, ρc]=δ_α, _1cρc^_1cb-δ_α,_1W_ρρc.






Next we are going to describe the action of sl_a(ℂ) as a Lie subalgebra of ^1(A) in ^2(A).


Let A be a toupie algebra with a>0, and let 

    E={ρα^(k): ρ is a relation from  0  to ω and α^(k)∈_0ℬ_ω-Z }
 
be a subset of the basis of ^2(A). 

The space ^2(A) decomposes, as representation of sl_a(ℂ), in a direct sum as follows:


    ^2(A)≃⊕_dim_ℂ <E> V_0  ⊕⊕_ρ:s(ρ)=0, t(ρ)=ω V
 
where V_0≃ℂ is the trivial representation and V is the standard representation of sl_a(ℂ).



Recall that D_2=0 and so Ker D_2=_E^e(kℛ, A). Observe that
^2(A) is generated by the classes of the elements of the set 
{ρα^(i):ρ is a relation from  0  to ω and α^(i)∈ Z}∪{ρα^(j):ρ is a relation from  0  to ω and α^(j)∈_0ℬ_ω-Z}.

Using Proposition <ref> and Proposition <ref>, we identify the generators of sl_a(ℂ) with the set 
{x_j, w_pq  with  p≠ q;  p,q=1,…, a  and  j=2,…, a} in ^1(A). Now identify the set 
    𝔥={b_1α^(1)α^(1)+b_2α^(2)α^(2)+… +b_aα^(a)α^(a): b_1+b_2+… +b_a=0}=⟨ x_j: j=2,…, a⟩
 
with the diagonal matrices with trace zero in sl_a(ℂ). Using the invariance of the Jordan decomposition 
–see Theorem 9.20 in <cit.>– we know that 𝔥 acts diagonally on ^2(A). 
This means that ^2(A)=⊕ V_λ where λ runs over 𝔥^* and V_λ={v∈ V: h.v=λ(h)v for all h∈𝔥}
 is the weight space of weight λ.
 For every i, 1≤ i≤ a, let L_i∈𝔥^* be defined by L_i(b_1α^(1)α^(1)+… +b_aα^(a))=b_i.
Notice that 𝔥^*=ℂ{L_1, …, L_a}/⟨ L_1+… +L_a⟩.

Given b_1α^(1)α^(1)+… +b_aα^(a)α^(a)∈𝔥, ρ a relation from 0 to ω and α^(i)∈ Z, 
and i such that 1≤ i ≤ a, the equality (<ref>) implies that

    [b_1α^(1)α^(1)+… +b_aα^(a)α^(a), ρα^(i)]=b_iρα^(i),


therefore ρα^(i) belongs to the weight space associated to the weight L_i.
Let us see that L_1 is a maximal weight of ^2(A). For that, again by (<ref>), it is enough to observe that given 
w_ij=α^(i)α^(j) with j< i, the bracket [w_ij, ρα^(1)] is zero. 
Using existence and uniqueness theorems –see <cit.> Chapter VI Theorems A and B– and the fact that L_1 is the unique maximal weight in the standard representation, 
we conclude that the irreducible representation generated by ρα^(1), that turns out to be 
⟨{ρα^(i):ρ is a relation from 0 to ω and α^(i)∈ Z}⟩, 
is isomorphic to the standard representation for each ρ from 0 to ω.

On the other hand, if α^(i)∈_0ℬ_ω-Z, then

    [b_1α^(1)α^(1)+… +b_aα^(a)α^(a), ρα^(i)]=0

and this implies that if ρα^(i) is not zero in ^2(A), 

then it belongs to the weight space associated to the weight 0. 
Besides, in this case, the subrepresentation generated by ρα^(i), which is ℂ.ρα^(i), is isomorphic to the trivial representation and we obtain the decomposition we were looking for.




Given a relation ρ from 0 to ω, the k-vector space V_ρ=⟨ρα^(k):α^(k)∈_0ℬ_ω⟩ is a Lie subrepresentation of ^1(A) in ^2(A).



Let A be a toupie algebra such that a is positive. The decomposition of ^2(A) as Lie representation of ^1(A) is

    ^2(A)=⊕_ρ:s(ρ)=0, t(ρ)=ωV_ρ,

where V_ρ=⟨ρα^(k):α^(k)∈_0ℬ_ω⟩. Besides, for every relation ρ from 0 to ω, the ^1(A)-module V_ρ 
is indecomposable.
Moreover,
if D=a, then V_ρ is irreducible for every ρ. The converse holds if there exists a monomial relation from 0 to ω.



It is clear that ^2(A)=∑_ρ:s(ρ)=0, t(ρ)=ωV_ρ and that V_ρ∩Σ_ρ'≠ρV_ρ'={0} for every ρ from 0 
to ω.
Let us now see that V_ρ is indecomposable for every relation ρ from 0 to ω.
We know that, since a is positive, there exists at least one arrow from 0 to ω which will be called α^(1). We assert that ρα^(1) is not zero 
in ^2(A) since ρα^(1) does not belong to Im(D_1). In the particular case where D=1, we have dim V_ρ=1 
and V_ρ es indecomposable.
In case D>1, given α^(i)∈_0ℬ_ω with i≠ 1:

    [α^(1)α^(i), ρα^(1)]=ρα^(i).

This equality implies that the orbit of the action of ^1(A) on ρα^(1) is V_ρ and then V_ρ is indecomposable.

We will next prove that if D=a, then V_ρ is irreducible for every relation ρ from 0 to ω.
Since D=a, using Theorem <ref>, we know that ^1(A)=A”_1⊕ sl_a(ℂ). Let us call V_ρ the representation of the Lie 
subalgebra sl_a(ℂ) with underlying vector space V_ρ. Using Theorem <ref>, there is an isomorphism V_ρ≃ V where V is the 
standard representation of sl_a(C) and as we already know, it is irreducible. It remains to prove that V_ρ is also irreducible. Any non trivial subrepresentation 
of ^2(A) contained in V_ρ would also be a non zero subrepresentation of sl_a(ℂ) contained in V_ρ and that is absurd.
Finally, we will prove that if there exists a monomial relation ρ from 0 to ω and V_ρ is irreducible, then D=a.
Suppose that D≠ a. In this case there exists α^(i)∈_0ℬ_ω-Z such that ρα^(i) is not zero in ^2(A).
Observe that H:=⟨ρα^(i)⟩ is a representation of ^1(A) contained in V_ρ, since given αc∈^1(A), 
Eq. (<ref>) implies that,


    [αc, ρα^(i)]=δ_α, _1(α^(i)) ρα^(i)-δ_α,_1ρρα^(i),

but a is positive and V_ρ is irreducible so this leads us to a contradiction and we conclude that D=a.






 §.§ ^i(A), i≥ 3 as representation of ^1(A)




Fix i>2. Given α∈ Q_1 and b a non zero path in A such that 0=s(α)=s(b) and t(α)=t(b),
consider αb∈^1(A). Also, 
given an (i-1)-ambiguity u from 0 to ω and a path c in _0ℬ_ω consider 
that, as usual,  we will identify with its class uc∈^i(A). 
Next we will compute the Gerstenhaber bracket of these two elements. Given an (i-1)-ambiguity w,

    [αb, uc](w)   =φ_i^*[η_1^*(αb), η_i^*(uc)](w)
       =[η_1^*(αb), η_i^*(uc)]φ_i(1⊗ w⊗ 1)
       =η_1^*(αb)∘η_i^*(uc)(φ_i(1⊗ w⊗ 1))-η_i^*(uc)∘η_1^*(αb)(φ_i(1⊗ w⊗ 1)).


The first term is equal to

    η_1^*(αb)(η_i^*(uc)(φ_i(1⊗ w⊗ 1)))   =η_1^*(αb)((uc)η_i(φ_i(1⊗ w⊗ 1))
       =η_1^*(αb)(uc)(1⊗ w⊗ 1)
       =δ_u,wη_1^*(αb)(c)
       =δ_u,w(αb)∑_cc^(1)⊗c^(2)⊗ c^(3)
       ={[ δ_w,uc^αb,  if α=_1c;;        0 , otherwise. ].


Observe that in the third equality we have used Proposition <ref>.
Let us compute the second term. By Proposition <ref>, we know that φ_i(1⊗ w⊗ 1) has the following form,

    ∑_a^(1)… a^(i+1)=w1⊗a^(1)⊗…⊗a^(i)⊗ a^(i+1).

Thus,

η_i^*(uc)∘η_1^*(αb)(φ_i(1⊗ w⊗ 1))=∑_a^(1)… a^(i+1)=wη_i^*(uc)∘η_1^*(αb)(1⊗a^(1)⊗…⊗a^(i)⊗ a^(i+1))=∑_a^(1)… a^(i+1)=wη_i^*(uc)∑_j=1^i((a^(1)⊗…⊗η_1^*(αb)(a^(j))⊗…⊗a^(i)) a^(i+1)).

Since α is the first arrow of some branch, we know that η_1^*(αb)(a^(j))=0 for j≠ 1, so the last expression equals 

    ∑_a^(1)… a^(i+1)=wη_i^*(uc)((η_1^*(αb)(a^(1))⊗…⊗a^(i)) a^(i+1))
 
which, by definition of η^∗ and using again that α is the first arrow of some branch, is

    δ_α, _1ω∑_a^(1)… a^(i+1)=wη_i^*(uc)(( a^(1)⊗…⊗a^(i)) a^(i+1)).

By Proposition <ref> we have that,

    δ_α, _1ω∑_a^(1)… a^(i+1)=wη_i^*(uc)(( a^(1)⊗…⊗a^(i)) a^(i+1))   =δ_α, _1ω(uc)η_i(∑_a^(1)… a^(i+1)=w (1⊗ a^(1)⊗…⊗a^(i)⊗ a^(i+1)))
       =δ_α, _1ω(uc)(1⊗ w⊗ 1)
       =δ_u,wδ_α, _1ωc.


Summarising, given αb∈^1(A) and uc∈^i(A):

    [αb, uc]=δ__1c,αuc^αb-δ__1u,αuc.


Theorems <ref> and <ref> can be adapted to the general case, that is, i≥ 2. 
We state them now and we omit the proofs since they are analogous to the case i=2.


Consider a toupie algebra A with a> 0. The decomposition of ^i(A) as a representation of sl_a(ℂ) is:

    ^i(A)≃⊕_i=1^D-a V_0  ⊕⊕_u:s(u)=0, t(u)=ω V,

where V_0≃ℂ is the trivial representation and V is the standard representation of sl_a(ℂ).


Given an (i-1)-ambiguity u from 0 to ω, the k-vector subspace of 
^i(A) V_u=⟨ uα^(k)  |  α^(k)∈_0ℬ_ω⟩ is a Lie subrepresentation of ^1(A).


Let A be a toupie algebra with a>0. The decomposition of ^i(A) as a Lie representation of  ^1(A) is the following:

    ^i(A)=⊕_u:s(u)=0, t(u)=ωV_u

where V_u=⟨ uα^(k)  |  α^(k)∈_0ℬ_ω⟩. Besides, note that for every (i-1)-ambiguity u from 0 to ω, 
the module V_u is indecomposable.
The representation V_u is irreducible if and only if D=a.


We end this article with an example, for which we compute the whole structure.




Consider the toupie algebra A=kQ/I where Q is the quiver bellow with #Q_0=13 and #Q_1=15:





Let ρ'_1=α_0^5α_1^5-α_0^6α_1^6 and σ_i=α_i^4α_i+1^4α_i+2^4α_i+3^4 with i=0, …, 4
and I=⟨ρ'_1, σ_i: i=0, … ,4⟩.
Note that there are two arrows from 0 to ω, there is one branch  not involved in any relation, one branch  with monomial relations and two branches involved in a non 
monomial relation. This means that D=4, a=2, l=1, m=1 and n=2.

There are four 2-ambiguities, α_0^4α_1^4α_2^4α_3^4α_4^4, α_1^4α_2^4α_3^4α_4^4α_5^4, 
α_2^4α_3^4α_4^4α_5^4α_6^4  and α_3^4α_4^4α_5^4α_6^4α_7^4 and one 3-ambiguity α^(4), for j≥ 4 the set of j-ambiguities is empty.

The Hochschild cohomology is the following: 


  * HH^0(A)=⟨∑_i∈ Q_0e_i e_i⟩



  * HH^1(A) we obtain that
HH^1(A)=⟨ y_4, w_12, w_21, x_2, z_13, z_23, z_16, z_26, t_1, t_2⟩, where we have used the previously defined notation.

  * HH^2(A)=⟨ρ'_1α^(1), ρ'_1α^(2), ρ'_1α^(3)⟩ 

  * HH^3(A)=0 

  * HH^4(A)=⟨α^(4)α^(1), α^(4)α^(2), α^(4)α^(3), α^(4)α^(6)⟩

  * HH^i(A)=0 for i>4.

The decomposition of the Lie algebra HH^1(A) is the following:

    HH^1(A)≅⟨ y_4⟩⊕ sl_2(ℂ)⋉ (⟨ t_1, t_2⟩⋉⟨ z_13, z_23, z_16, z_26⟩ )


The non-null Gerstenhaber brackets of HH^1(A) with HH^2(A) are the following:
 3
 
  * [w_12,ρ'_1α^(1) ]=ρ'_1α^(2)

  * [x_2,ρ'_1α^(1)]=-ρ'_1α^(1)

  * [z_13, ρ'_1α^(1)]=ρ'_1α^(3)

  * [t_2, ρ'_1α^(1)]=-ρ'_1α^(1)

  * [w_21,ρ'_1α^(2)]=ρ'_1α^(1)

  * [x_2, ρ'_1α^(2)]=ρ'_1α^(2)

  * [z_23, ρ'_1α^(2)]=ρ'_1α^(3)

  * [t_2, ρ'_1α^(2)]=-ρ'_1α^(2)

  * [t_1, ρ'_1α^(3)]=ρ'_1α^(3)

  * [t_2, ρ'_1α^(3)]=-ρ'_1α^(3)

The non-null Gerstenhaber brackets of HH^1(A) with HH^4(A) are the following:
3

  * [y_4, α^(4)α^(1)]=-α^(4)α^(1)

  * [w_12, α^(4)α^(1)]=α^(4)α^(2)

  * [x_2, α^(4)α^(1)]=-α^(4)α^(1)

  * [z_13, α^(4)α^(1)]=α^(4)α^(3)

  * [z_16, α^(4)α^(1)]=α^(4)α^(6)

  * [y_4, α^(4)α^(2)]=-α^(4)α^(2)

  * [w_21, α^(4)α^(2)]=α^(4)α^(1)

  * [x_2, α^(4)α^(2)]=α^(4)α^(2)

  * [z_23, α^(4)α^(2)]=α^(4)α^(3)

  * [z_26, α^(4)α^(2)]=α^(4)α^(6)

  * [y_4,α^(4)α^(3)]=-α^(4)α^(3)

  * [t_1, α^(4)α^(3)]=α^(4)α^(3)

  * [y_4,α^(4)α^(6)]=-α^(4)α^(6)
 
  * [t_2, α^(4)α^(6)]=α^(4)α^(6)






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