A formality framework for commutative deformations
Olivier Elchinger


Gain and loss  in open quantum systems
    
Hichem Eleuch^1[email: hichemeleuch@tamu.edu] and 
Ingrid Rotter^2[email: rotter@pks.mpg.de, corresponding author]
    December 30, 2023
=======================================================================================================================







In this article, we use Harrison cohomology to provide a framework for commutative deformations. In particular, Kontsevich's result that formality of (the Hochschild complex of) an associative algebra implies its deformability is adapted for commutative algebras, with the Harrison complex.


Keywords: formality, Harrison cohomology, commutative deformations, eulerian idempotents

2010 AMS Subject Classification: 13D03, 13D10, 16T10



§ INTRODUCTION


Kontsevich showed in <cit.> the existence of an associative deformation quantization for the general case of smooth Poisson manifolds. He deduced this result from his general “formality statement”. Endowed with the Gerstenhaber bracket, the continuous Hochschild complex of the algebra A = 𝒞^∞(M) of smooth functions over a Poisson manifold admits a graded Lie algebraic structure, which controls the deformations of the associative commutative algebra A. Kontsevich shows that this complex is linked with its cohomology – which therefore controls the same deformations – by a L_∞-quasi-isomorphism, called a formality map.

Considering formality, the case of smooth manifolds is thus rather well understood using continuous Hochschild cohomology, and it is this tool which gives a lot of information about deformability (obstructions, rigidity,…). Moreover, if
the Hochschild complex of an associative algebra is formal in Kontsevich's sense, this algebra admits a quantization by deformation, but the converse does not hold, for example in the case of free algebras, see <cit.>.

Since formality methods work well to give complete answers to the deformation quantization in the regular case (both 𝒞^∞ and algebraic) it seems to be interesting – as proposed by Frønsdal
and Kontsevich in <cit.> – to look at the deformation quantization problem for more general singular Poisson manifolds.
The main problem is the fact that the HKR result of a “simple” Hochschild cohomology of the algebra of functions, generated by derivations, no longer holds, for example there may be non-trivial 2-cocycles which are symmetric. These symmetric cocycles are infinitesimal commutative deformations of the algebra of functions.
In order to systematically investigate commutative associative algebras, Harrison (<cit.>) described combinatorially the “commutative component” of the Hochschild complex, and proved that its cohomology is reduced to derivations if and only if the algebra is “regular”.




The main goal of this work is to adapt the result that formality implies deformation to the case of a commutative algebra, replacing Hochschild complex by Harrison complex.

I am grateful to Prof. Bordemann for his help and useful remarks.

In <ref> we recall Hochschild and Harrison (co)homology. <ref> and <ref> introduce tools coming from Hopf algebra theory: (co)freeness, convolution products, eulerian idempotents. This gives two descriptions of the Harrison complex, providing a short proof of a result of Barr. Finally, <ref> presents commutative deformations, and the aforementioned result <ref>. 






Let  be a field containing the rationals.



§ HOCHSCHILD AND HARRISON (CO)HOMOLOGY
 

Let A be a commutative -algebra, and consider its Hochschild complex _(A) with _n(A) = A ⊗ A^⊗ n.


Loday recalls in <cit.> the action of the symmetric group 𝔖_n on _n(A)

    𝔖_n ↷_n(A)    →_n(A) 
    σ.(a_0,a_1,…,a_n)     = (a_0,a_σ^-1(1),…,a_σ^-1(n))

as well as the shuffle product

    sh_p,q : _p(A) ×_q(A)    →_p+q(A) 
    
    (a_0,a_1,…,a_p) ∙ (a'_0,a_p+1,…,a_p+q)     = ∑_σ∈ Sh_p,qsgn(σ) σ.(a_0 a'_0,a_1,…,a_p+q)

where Sh_p,q are the (p,q)-shuffles, elements σ of 𝔖_p+q such that σ(1)<…<σ(p) and σ(p+1)<…<σ(p+q) ; and he also defines the shuffle map

    sh = ∑_p+q = n 
     p⩾ 1,q⩾ 1sh_p,q : _n(A) →_n(A)   as the action of the element   sh = ∑_σ∈ Sh_p,q
     p+q = n 
     p⩾ 1,q⩾ 1sgn(σ) σ∈[𝔖_n]


Endowed with the shuffle product (often noted ∙), the Hochschild complex is a commutative differential graded algebra augmented over A. Let I=⊕_n>0_n(A) be the augmentation ideal. The quotient (A) =  (A)/I^∙ 2 is a well defined complex since the Hochschild boundary map is a graded derivation for the shuffle product.

For any A-module M, Hochschild homology and cohomology are given by _(A,M) = H(_(A)⊗_A M) and ^(A,M) = H(_A(_(A),M)). The Harrison homology and cohomology are defined as _(A,M) = H(_(A)⊗_A M) and ^(A,M) = H(_A(_(A),M)).

Barr already proved in <cit.> that there are maps _(A,M) ↠_(A,M) and ^(A,M) ↪^(A,M).



§ TENSORIAL BIALGEBRAS
 

Let V be a -vector space. The tensorial module over V is given by V = ⊕_n ∈ V^⊗ n.



 §.§ Freeness and cofreeness


Endowed with the multiplication μ of concatenation, ( V,μ,) is the free associative algebra over V, characterized (up to isomorphism) by the universal property that each morphism ϕ from V to an associative algebra (A,μ_A) factors through V in ϕ = _V ∘ϕ.

Endowed with the comultiplication Δ of deconcatenation, ( V,Δ,ε) is the cofree coassociative conilpotent coalgebra over V, characterized (up to isomorphism) by the universal property that each morphism ϕ from a coaugmented conilpotent coalgebra (C,Δ_C) to V, satisfying ϕ(1)=0, factors through V in ϕ = _V ∘ϕ.

Likewise, for any linear map d : V → A, there exists a unique graded derivation along ϕ noted d :  V → A such that d|_V = d

; and for any linear map d : C^+ → V (with C^+ = ε_C), there exists a unique graded coderivation along ϕ noted d : C → V such that _V ∘d = d.




    [>=angle 90,baseline=(current bounding box.center)]
     (m) [matrix of math nodes,nodes in empty cells,column sep=2em,row sep=2em,text height=1.5ex,text depth=0.25ex,ampersand replacement=&]
    ( V,μ) && (A,μ_A) 
    & V 
    ;
    [->]
    (m-1-1) edge node [above] ϕ,d (m-1-3);
    [right hook->]
    (m-2-2) edge (m-1-1);
    [->]
    (m-2-2) edge node [below right] ϕ,d (m-1-3);
          [>=angle 90,baseline=(current bounding box.center)]
     (m) [matrix of math nodes,nodes in empty cells,column sep=2em,row sep=2em,text height=1.5ex,text depth=0.25ex,ampersand replacement=&]
    ( V,Δ) && (C,Δ_C) 
    & V 
    ;
    [->]
    (m-1-3) edge node [above] ϕ,d (m-1-1);
    [->>]
    (m-1-1) edge (m-2-2);
    [->]
    (m-1-3) edge node [below right] ϕ,d (m-2-2);
    
    d∘μ = μ_A ∘ (d⊗ϕ + ϕ⊗d)
          Δ∘d = (d⊗ϕ + ϕ⊗d) ∘Δ_C


More details on these structures can be found in <cit.>. The emphasis is put on the following formulas using convolutions products. For both the algebra and coalgebra setting the formulas are the same, only the convolution products changes. For more detailed proofs, see <cit.>.

The algebra morphism ϕ induced by ϕ : V → A is computed as ϕ = ∑_n ∈ϕ^⋆ n, the geometric serie using the convolution product ⋆ with respect to the multiplication μ_A and the comultiplication of deconcatenation Δ. The derivation d along ϕ induced by d and ϕ can be computed as d = ϕ⋆ d ⋆ϕ.

The coalgebra morphism ϕ coinduced by ϕ : C^+ → V is computed as ϕ = ∑_n ∈ϕ^⋆ n, the geometric serie using the convolution product ⋆ with respect to the multiplication of concatenation μ and the comultiplication Δ_C. The coderivation d along ϕ coinduced by d and ϕ can be computed as d = ϕ⋆ d ⋆ϕ. 


Moreover, ( V,μ,Δ_sh,,ε) is a bialgebra, Δ_sh being the morphism of associative algebras Δ_sh : ( V,μ) → ( V ⊗ V,μ^[2]) induced by _V ⊗ + ⊗_V.

Also, ( V,μ_sh,Δ,,ε) is a bialgebra, μ_sh being the morphism of coassociative coalgebras μ_sh : ( V ⊗ V,Δ^[2]) → ( V,Δ) coinduced by _V ⊗ε + ε⊗_V.



The shuffle product can also be seen as the commutative product resulting on the quotient V =  V/(x ⊗ y - (-1)^|x| |y| y ⊗ x,   x,y ∈ V). Since Δ_sh is cocommutative, it factors through the quotient, and thus ( V,μ_sh,Δ_sh,,ε) is also a bialgebra.




 §.§ Toolbox on operations


In this section, we present some relations between product, composition, convolution and counit which will be used later. Let α : V → V (or α_i) be a linear map.

Since V ⊗≅ V, we have that μ_sh∘ (α⊗ε) = α⊗ε are the same map from V ⊗ V → V since they send a ⊗ b ↦α(a)ε(b).

Let ϕ :  V ⊗ V → V be a coalgebra morphism, meaning that Δ∘ϕ = (ϕ⊗ϕ) ∘Δ^[2], with Δ^[2] = (id ⊗τ⊗ id) ∘ (Δ⊗Δ), that is Δ^[2] = perm∘ (Δ⊗Δ), the coproduct on each factor followed by a permutation so that the morphism is applied to the right elements.
We have

    (α_1 ⋆α_2) ∘ϕ   = μ_sh∘ (α_1 ⊗α_2) ∘Δ∘ϕ
       = μ_sh∘ (α_1 ⊗α_2) ∘ (ϕ⊗ϕ) ∘Δ^[2]
       = μ_sh∘ (α_1 ∘ϕ⊗α_2 ∘ϕ) ∘Δ^[2] = (α_1 ∘ϕ) ⋆_2 (α_2 ∘ϕ),

with the convolution product _⋆_2 _ = μ_sh∘ (_⊗_)∘Δ^[2].
The property of coalgebra morphism also reads Δ^(n-1)∘ϕ = ϕ^⊗ n∘Δ^[n], where Δ^(n-1) :  V → V^⊗ n is the (n-1)-fold of the associative coproduct, and Δ^[n] = perm∘ (Δ^(n-1)⊗Δ^(n-1)). We have

    (α_1 ⋆…⋆α_n) ∘ϕ = (α_1 ∘ϕ) ⋆_n …⋆_n (α_n ⊗ϕ)     with  _⋆_…_⋆_ = μ_sh∘ (_⊗_…_⊗_) ∘Δ^(n-1)
    _⋆_n_…_⋆_n_ = μ_sh∘ (_⊗_…_⊗_) ∘Δ^[n]

and we will write collectively  for those second kind of convolutions. Taking α_i = α and summing the previous equalities gives

    e^⋆α∘ϕ = e^(α∘ϕ).


Using this with ϕ = id ⊗ε, which indeed is a coalgebra morphism, we obtain

    (α_1 ⋆α_2) ⊗ε = (α_1 ⋆α_2) ∘ϕ = (α_1 ∘ϕ)  (α_2 ∘ϕ) = (α_1 ⊗ε)  (α_2 ⊗ε)
    and thus
    e^⋆α⊗ε = e^(α⊗ε).




§ EULERIAN IDEMPOTENTS
 

Following Loday and Vallette <cit.> and <cit.>, we define the eulerian idempotents on the commutative Hopf algebra ( V,μ_sh,Δ,,ε). We consider its convolution algebra (( V, V),⋆,ε), where the convolution product is _⋆_ = μ_sh∘ (_⊗_)∘Δ. We write id = ε + J so that J is the identity on V^⊗ n except for n = 0 on which it is 0. We define

    e^(1)log^⋆(id) = log^⋆(ε + J) = ∑_n ⩾ 1 (-1)^n+1J^⋆ n/n


In weight n we get that e^(1) : V^⊗ n→ V^⊗ n is given by e^(1)(x_1 … x_n) = e_n^(1)· (x_1 … x_n) for some uniquely defined element e_n^(1)∈[𝔖_n]. These elements are called the first eulerian idempotents. For i ⩾ 1, we define

    e^(i)(e^(1))^⋆ i/i!


Loday shows <cit.> that the elements e_n^(i)∈[𝔖_n] are orthogonal idempotents.

In low dimensions, the eulerian idempotents are

    n=1         e_1^(1) = id; 
    
    n=2         e_2^(1) = 1/2(id + (1 2)),    e_2^(2) = 1/2(id - (1 2)); 
    
    n=3         e_3^(1) = 1/3id - 1/6((1 2 3)+(1 3 2)-(1 2)-(2 3)) - 1/3(1 3), 
        e_3^(2) = 1/2(id+(1 3)),    e_3^(3) = 1/6(id+(1 2 3)+(1 3 2)-(1 2)-(2 3)-(1 3)).



The first eulerian idempotent e^(1) is a derivation for μ_sh along ε, 
    e^(1)∘μ_sh = μ_sh∘ (e^(1)⊗ε + ε⊗ e^(1)).




Writing μ_sh = id ∘μ_sh = e^⋆ e^(1)∘μ_sh = e^(e^(1)∘μ_sh), we will show that e^(e^(1)⊗ε + ε⊗ e^(1)) = e^(e^(1)∘μ_sh), which gives the result since e^(1)⊗ε + ε⊗ e^(1) = μ_sh∘ (e^(1)⊗ε + ε⊗ e^(1)).

We have e^(A+B) = e^ A e^ B, providing A  B = B  A. Set A =α⊗ε and B = ε⊗α, with α a linear map. Let a,b ∈ V, we use Sweedler notation Δ(a) = ∑_(a) a^(1)⊗ a^(2). Note that |a| = |a^(1)| + |a^(2)| for any elements a^(1),a^(2) of the sum. We have

    ( α⊗εε⊗α)(a ⊗ b)    = μ_sh∘ (α⊗ε⊗ε⊗α) ∘Δ^[2](a ⊗ b) 
       = μ_sh(∑_(a),(b) (-1)^|a^(2)| |b^(1)|α(a^(1)) ε(b^(1)) ⊗ε(a^(2)) α(b^(2)) ) 
       = μ_sh(∑_(a),(b)α(a^(1)ε(a^(2))) ⊗α(b^(2)ε(b^(1))) ) 
        = α(a) ∙α(b)

since terms with elements a^(2) or b^(1) of degree different from zero are killed by the counity; and since |a^(2)| = |a| - |a^(1)|, |b^(1)| = |b| - |b^(2)|,

    ( ε⊗αα⊗ε)(a ⊗ b)    = μ_sh∘ (ε⊗α⊗α⊗ε) ∘Δ^[2](a ⊗ b) 
       = μ_sh(∑_(a),(b) (-1)^|a^(2)| |b^(1)|ε(a^(1)) α(b^(1)) ⊗α(a^(2)) ε(b^(2)) ) 
       = (-1)^|a| |b|α(b) ∙α(a) = α(a) ∙α(b)

Taking α = e^(1), we thus have

    e^(e^(1)⊗ε + ε⊗ e^(1))   = e^(e^(1)⊗ε) e^(ε⊗ e^(1)) = ( e^⋆ e^(1)⊗ε) ( ε⊗ e^⋆ e^(1)) 
       = (id ⊗ε)  (ε⊗ id) = μ_sh∘ (id ⊗ε⊗ε⊗ id) ∘Δ^[2] = μ_sh∘ (id ⊗ id) = μ_sh = e^(e^(1)∘μ_sh).



This proposition implies the equivalence of the two original definitions of Harrison (co)homology of a commutative algebra A as given by Harrison <cit.> and Barr <cit.>.


The complexes (A) = (A)/I^∙ 2 and e^(1)(A) are isomorphic.



Let a,b ∈(A). Since e^(1) is a derivation of μ_sh along ε, we have

    e^(1)(a ∙ b) = e^(1)(a) ε(b) + ε(a) e^(1)(b).

If a,b ∈ I are elements in the ideal of augmentation, then ε(a) = 0 = ε(b), thus e^(1)(a ∙ b) = 0, hence I ∙ I ⊂(e^(1)). Also (Sweedler's summations implied)

    e^(1)(a) = ∑_n ⩾ 1 (-1)^n+1id^⋆/n(a) = a-1/2 a^(1)∙ a^(2) + 1/3 a^(1)∙ a^(2)∙ a^(3) - …

so a-e^(1)(a) ∈ I ∙ I, hence (id - e^(1)) = (e^(1)) ⊂ I ∙ I.

So we have the decomposition

    (A) = (id-e^(1))(A) ⊕ e^(1)(A) = I^∙ 2⊕ e^(1)(A)

which gives the result.


Barr's proof of <cit.> consists in a construction by induction of a sequence e_n ∈[𝔖_n] of idempotent maps commuting with the Hochschild boundary map and leaving the shuffle products sh_p,q invariant. Here this proof use the property of the whole map e^(1) of being a graded derivation. Note that e^(1)_n = id - e_n in Barr's notation.


In particular, this gives two descriptions of Harrison cochains:

    (A,M) = ((A)/I^∙ 2,M) = (e^(1)(A),M)

they can be viewed as maps A → A that cancel on shuffles, or invariant by the first eulerian idempotent.


The second Harrison module ^2(A,M) = {f : A^⊗ 2→ M, f(a,b)=f(b,a)} consists of symmetric maps.




§ COMMUTATIVE DEFORMATIONS
 

Let (A,μ_0) be a commutative -algebra. In <cit.>, Frønsdal defines commutative deformations. A formal, abelian *-product on A is a commutative, associative product on the space A[[λ]] of formal power series in the formal parameter λ with coefficients in A, given by formal series

    f * g = ∑_n ∈λ^n μ_n(f,g).


Associativity for * is the condition (f * g) * h = f * (g * h) or equivalently A_n(f,g,h) = 0 for all n ∈, where A_n is the associator of order n for *

    A_n(f,g,h) ∑_k=0^n (μ_k(μ_n-k(f,g),h) - μ_k(f,μ_n-k(g,h))).


For any product μ, its associator A(a,b,c) μ(μ(a,b),c)-μ(a,μ(b,c)) satisfies

    0 = [μ,A]_G(a,b,c,d) = μ(A(a,b,c),d) + μ(a,A(b,c,d)) - A(μ(a,b),c,d) + A(a,μ(b,c),d) - A(a,b,μ(c,d))

with [ , ]_G the Gerstenhaber bracket. 



Let A = ∑_n ∈λ^n A_n be the associator of a *-product *. Suppose that * is associative to order r ⩾ 1, A_0 = … = A_r = 0.
Equation (<ref>) at order λ^r+1 reads 0 = (δ A_r+1)(a,b,c,d), with δ the Hochschild coboundary, hence A_r+1 is a Hochschild 3-cocycle. Moreover A_r+1 = -δμ_r+1 + A_r+1', where A_r+1' is A_r+1 without the first and last term in the sum. This shows that A_r+1' is also a 3-cocycle, and A_r+1 = 0 ⇔ A_r+1' = δμ_r+1, so that * is associative to order r+1 is equivalent to A_r+1' being a 3-coboundary.


This proves that the obstruction to promote associativity from order r to order r+1 are in ^3(A,A). Moreover, if μ_i, 1 ⩽ i ⩽ r are symmetric, then a direct computation shows that A_r+1' is invariant by e^(1)_3, so the obstructions to extend a formal abelian *-product to higher orders are more precisely in ^3(A,A). 



Barr showed that Harrison cohomology is included in Hochschild cohomology, but it is already the case for the complexes as differential graded Lie algebras.

 
The Harrison complex of cochains ((A,A)[1],δ,[ , ]_G) is a differential graded sub-Lie algebra of the Hochschild complex ((A,A)[1],δ,[ , ]_G).


We first prove the following lemma, see also <cit.>.


Cochains of (A,A) induces derivations of ( A,μ_sh).




Let d : A^⊗ k→ A be a cochain in ^k(A,A) ⊂^k(A,A). It induces d, coderivation of ( A,Δ). We want to show that it is also a derivation for μ_sh, 
    d∘μ_sh = μ_sh∘ (d⊗ id + id ⊗d).

Since μ_sh : ( A ⊗ A, Δ^[2]) → ( A,Δ) is a coalgebra morphism, both sides of the equation are coderivation from ( A ⊗ A,Δ^[2]) to ( A,Δ) along μ_sh.

Projecting on A, we have on the left-hand side d ∘μ_sh(a ⊗ b) = d(a ∙ b) and on the right-hand side (_A ⊗ε + ε⊗_A) ∘ (d⊗ id + id ⊗d)(a ⊗ b) = (d ⊗ε + ε⊗ d)(a ⊗ b) = d(a) ε(b) + ε(a) d(b) because ε∘d = 0. But since d vanishes on I^∙ 2 and ε on I, the two expressions are equal for all a,b ∈ A. Since the left and right-hand side are coderivations along μ_sh having the same projection, they must be equal by unicity.



Let f,g ∈(A,A)[1]. Using the previous lemma, we have

    [f,g]_G ∘μ_sh   = f ∘g∘μ_sh + (-1)^|f| |g| g ∘f∘μ_sh
       = f ∘μ_sh∘ (g⊗ id + id ⊗g) + (-1)^|f| |g| g ∘μ_sh∘ (f⊗ id + id ⊗f)
    thus
    [f,g]_G(a ∙ b)    = f (g(a) ∙ b ± a ∙g(b) ) + (-1)^|f| |g| g ( f(a) ∙ b ± a ∙f(b) ) = 0

hence the vanishing of f and g on I^∙ 2 imply the one of [f,g]_G on I^∙ 2, so (A,A)[1] is closed for the Gerstenhaber bracket.


We recall Kontsevich's notion of formality. For better readability, we note = (A,A) and = (A,A) the Hochschild complex and cohomology of A.

 
The complex  is called formal if there is a L_∞-quasi-isomorphism Φ : ([2]) →([2]) (morphism of differential graded coalgebras of degree 0), 
    (Φ⊗Φ) ∘Δ_[2] = Δ_[2]∘Φ    and    b+D∘Φ = Φ∘d,

such that the restriction Φ_1 of Φ to H[2] is a section. The map Φ is called a formality map.


Here b  [μ_0, ]_G is the same as the Hochschild coboundary δ up to a global sign. We recall that the projection of the Gerstenhaber bracket gives a graded Lie bracket [ , ]_s on the shifted cohomology space [1]. The maps D  [ , ]_G[1] and d  [ , ]_s[1] denote the shifted brackets, which are symmetric; D and d are the induced coderivations on ([2]) and ([2]).

By extension, we will say that an associative algebra A is formal if it is the case for its Hochschild complex . For a commutative algebra A, we keep the same definition of formality, but now taking = (A,A) and = (A,A) the Harrison complex and cohomology of A.


 
(commutative) formality ⇒ (commutative) déformation



For associative algebras, the result goes back to Kontsevich <cit.>, with the given framework, it adapts well to commutative algebras. We follow here the presentation of <cit.>.
Let π∈^2[[λ]] = [2]^0[[λ]]. We want to construct a formal associative (commutative) deformation μ=μ_0+μ_* where μ_* ∑_r=1^∞λ^r μ_r such that the cohomology class [μ_1] of μ_1 is equal to π. A necessary condition for this is

    [π,π]_s = 0

so we suppose the chosen element π satisfies it.


Consider ([2])[[λ]] and ([2])[[λ]] as topological bialgebras (with respect to the λ-adic topology) with the canonical extension of all the structure maps. Note that the tensor product is no longer algebraic, but given by (([2]) ⊗([2]))[[λ]]. For a general graded vector space V it can be easily seen that the group-like elements of V[[λ]] are no longer exclusively given by , but by exponential functions of any primitive elements of degree zero, they take the form e^∙λ v with v∈ V^0[[λ]]. The image Φ(e^∙λπ) of the grouplike element e^∙λπ in ([2])[[λ]] under the formality map Φ is a grouplike element in ([2])[[λ]] and thus takes the form e^∙μ_* with μ_* ∈λ^2[[λ]]. Since [π,π]_s = 0 it follows that d(e^∙λπ)=0, and therefore (b+D)(e^∙λμ_*) = 0. Projecting this last equation to [2]^0[[λ]] = ^2[[λ]], we get the Maurer-Cartan Equation

    0 = bμ_* + 1/2 [μ_*,μ_*]_G = 1/2 [μ_0 + μ_*,μ_0 + μ_*]_G,

showing the associativity of μ = μ_0 + μ_*. Hence μμ_0 + μ_* is a formal associative deformation of the algebra (A,μ_0).
In the commutative case with = (A,A), μ_* ∈λ^2[[λ]] is equivalent to the commutativity of μ_i for i⩾ 1, so the resulting product μ is commutative.



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