Gerstenhaber Algebra Research Articles

Poincaré duality for smooth Poisson algebras and BV structure on Poisson cohomology

Similar to the modular vector fields in Poisson geometry, modular derivations are defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson module with the modular derivation, the Poisson cochain complex with values in any Poisson module is proved to be isomorphic to the Poisson chain complex with values in the corresponding twisted Poisson module. Then a version of twisted Poincaré duality is proved between the Poisson homologies and cohomologies. Furthermore, a notion of pseudo-unimodular Poisson structure is defined. It is proved that the Poisson cohomology as a Gerstenhaber algebra admits a Batalin-Vilkovisky operator inherited from some one of its Poisson cochain complex if and only if the Poisson structure is pseudo-unimodular. This generalizes the geometric version due to P. Xu. The modular derivation and Batalin-Vilkovisky operator are also described by using the dual basis of the Kähler differential module.

Hochschild cohomology of generalised Grassmannians

We compute the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of generalised Grassmannians, i.e., partial flag varieties associated to maximal parabolic subgroups in a simple algebraic group, in terms of representation-theoretic data.We explain how the decomposition is concentrated in global sections for the (co)minuscule and (co)adjoint generalised Grassmannians, and conjecture that for (almost) all other cases the same vanishing of the higher cohomology does not hold. Our methods give an explicit partial description of the Gerstenhaber algebra structure for the Hochschild cohomology of cominuscule and adjoint generalised Grassmannians. We also consider the case of adjoint partial flag varieties in type A, which are associated to certain submaximal parabolic subgroups.

Operations on the Hochschild bicomplex of a diagram of algebras

A diagram of algebras is a functor valued in a category of associative algebras. I construct an operad acting on the Hochschild bicomplex of a diagram of algebras. Using this operad, I give a direct proof that the Hochschild cohomology of a diagram of algebras is a Gerstenhaber algebra. I also show that the total complex is an L∞-algebra. The same results are true for the reduced and asimplicial subcomplexes and asimplicial cohomology. This structure governs deformations of diagrams of algebras through the Maurer-Cartan equation.

Polyvector fields for Fano 3-folds

We compute the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of Fano 3-folds. This is the first step in understanding the non-trivial Gerstenhaber algebra structure of this invariant, and yields some initial insights in the classification of Poisson structures on Fano 3-folds of higher Picard rank.

Massey products for algebras over operads

We define a generalization of Massey products for algebras over a Koszul operad in characteristic zero, extending Massey’s and Allday’s and Retah’s in the associative and Lie cases, respectively. We establish connections with minimal models and with Dimitrova’s universal operadic cohomology class. We compute a Gerstenhaber algebra example and a hypercommutative algebra example related to the Chevalley–Eilenberg complex of the Heisenberg Lie algebra.

Formes différentielles non commutatives et Algèbres de Gerstenhaber

In this work we show that for any topological space X having the homotopy type of a CW-complex and for any commutative ring R, the singular cohomology H * (X;R) is a Gerstenhaber algebra (see. [7]). In fact we prove that H * (X;R) satisfies the conditions of a generalization of the Gerstenhaber algebras.

Gerstenhaber algebra of the Hochschild cohomology of an associative conformal algebra

Gerstenhaber algebra of the Hochschild cohomology of an associative conformal algebra

Gerstenhaber algebra of ℤ2 × ℤ2-Galois covering of quantum exterior algebra

Let [Formula: see text] be the quantum exterior algebras over a field [Formula: see text] with char [Formula: see text], [Formula: see text] be the [Formula: see text]-Galois coverings of [Formula: see text]. In this paper, a basis of each degree of Hochschild cohomology of [Formula: see text], the cup product and the Gerstenhaber bracket on Hochschild cohomology of [Formula: see text] are given.

Hochschild cohomology of twisted tensor products

The tensor product $$R\otimes S$$ of two algebras can have its multiplication deformed by a bicharacter to yield a twisted tensor product $$R\otimes ^t S$$ . We completely describe the Hochschild cohomology of $$R\otimes ^t S$$ in terms of the Hochschild cohomology of the components R and S, including the full Gerstenhaber algebra structure. This description generalizes a result of Bergh and Oppermann. A number of interesting classes of noncommutative algebras arise as bicharacter twisted tensor products, sometimes in non-obvious ways. The main result thereby allows us to significantly simplify various calculations in the literature, and to compute Hochschild cohomology in several new classes of examples. In particular, we fully compute the Hochschild cohomology of quantum complete intersection algebras, with any number of indeterminates. One new tool which goes into the main theorem is orbit Hochschild cohomology, which can be defined for algebras with a group action, and which satisfies twisted versions of the usual Gerstenhaber algebra axioms.

The cohomology rings of homogeneous spaces

Let G be a compact connected Lie group and K a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of G and K is invertible in a given principal ideal domain k. It is known that in this case the cohomology of the homogeneous space G / K with coefficients in k and the torsion product of H ∗ ( B K ) and k over H ∗ ( B G ) are isomorphic as k-modules. We show that this isomorphism is multiplicative and natural in the pair ( G , K ) provided that 2 is invertible in k. The proof uses homotopy Gerstenhaber algebras in an essential way. In particular, we show that the normalized singular cochains on the classifying space of a torus are formal as a homotopy Gerstenhaber algebra.

Polyvector fields and polydifferential operators associated with Lie pairs

We prove that the spaces \operatorname{tot} \big(\Gamma ({\Lambda^\bullet A^\vee}) \otimes_R{{\mathcal{T}_{poly}^{\bullet}}}\big) and \operatorname{tot} \big(\Gamma ({\Lambda^\bullet A^\vee}) \otimes_R{{\mathcal{D}_{poly}^{\bullet}}}\big) associated with a Lie pair (L,A) each carry an L_\infty algebra structure canonical up to an L_\infty isomorphism with the identity map as linear part. These two spaces serve, respectively, as replacements for the spaces of formal polyvector fields and formal polydifferential operators on the Lie pair (L,A) . Consequently, both \mathbb{H}^{\bullet}_{\operatorname{CE}}(A,{\mathcal{T}_{poly}^{\bullet}}) and \mathbb{H}^{\bullet}_{\operatorname{CE}}(A,{\mathcal{D}_{poly}^{\bullet}}) admit unique Gerstenhaber algebra structures. Our approach is based on homotopy transfer and the construction of a Fedosov dg Lie algebroid (i.e. a dg foliation on a Fedosov dg manifold).

Cocyclic complexes of Hopf algebras with special antipodes

We show that for any bimodule M and bicomodule C of a Hopf algebra H with involutive antipode, Hochschild complex and are cocyclic -modules. Their para-cocyclic operators are compatible in Gerstenhaber-Schack bicomplex of a Hopf algebra morphism which is a combination of Hochschild complex and coHochschild complex for any and make the bicomplex into a cylindrical -module under appropriate conditions. From this bicomplex, we get a cocyclic structure of the diagonal complex. With these cocyclic operators, the operadic structure of the diagonal complex is cyclic, so that the Gerstenhaber algebra structure on the diagonal cohomology evolves into a Batalin-Vilkovisky algebra. Hence we obtain the structure of Gerstenhaber-Schack cohomology of which is isomorphic to the diagonal cohomology. Our results can be applied to group algebra and arbitrary G-graded Hopf algebra A. Let be ϵA (resp. uB , idA ). Then we can get the Gerstenhaber structure on Hochschild cohomology (resp. Adams Cobar construction of B, Gerstenhaber-Schack cohomology of A). Moreover, the Gerstenhaber structure operators of ϵA and uB commute with composition.

Homotopy liftings and Hochschild cohomology of some twisted tensor products

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov’s homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.

Cup product on $A_\infty$-cohomology and deformations

We propose a simple method for constructing formal deformations of differential graded algebras in the category of minimal A_\infty -algebras. The basis for our approach is provided by the Gerstenhaber algebra structure on A_\infty -cohomology, which we define in terms of the brace operations. As an example, we construct a minimal A_\infty -algebra from the Weyl–Moyal \ast -product algebra of polynomial functions.

Homotopy Gerstenhaber formality of Davis–Januszkiewicz spaces

A homotopy Gerstenhaber structure on a differential graded algebra is essentially a family of operations defining a multiplication on its bar construction. We prove that the normalized singular cochain algebra of a Davis-Januszkiewicz space is formal as a homotopy Gerstenhaber algebra, for any coefficient ring. This generalizes a recent result by the author about classifying spaces of tori and also strengthens the well-known dga formality result for Davis-Januszkiewicz spaces due to the author and Notbohm-Ray. As an application, we determine the cohomology rings of free and based loop spaces of Davis-Januszkiewicz spaces.

Log smooth deformation theory via Gerstenhaber algebras

We construct a kleft[ !left[ Qright] !right] -linear predifferential graded Lie algebra L^{bullet }_{X_0/S_0} associated to a log smooth and saturated morphism f_0: X_0 rightarrow S_0 and prove that it controls the log smooth deformation functor. This provides a geometric interpretation of a construction in Chan et al. (Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties, 2019. arXiv:1902.11174) whereof L^{bullet }_{X_0/S_0} is a purely algebraic version. Our proof crucially relies on studying deformations of the Gerstenhaber algebra of polyvector fields; this method is closely related to recent developments in mirror symmetry.

The Hochschild Cohomology Ring of Temperley–Lieb Algebras Revisited

We determine the Gerstenhaber algebra structure on the Hochschild cohomology ring of Temperley–Lieb algebras in this paper.

Homotopy Gerstenhaber algebras are strongly homotopy commutative

We show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff-Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a cup-1 product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense of Munkholm.

Poisson cohomology of plane Poisson structures with isolated singularities revisited

Continuing the work of Monnier [Israel J. Math. 129, 189–207 (2002)], we determine the Gerstenhaber algebra structure over the Poisson cohomology groups for a large class of Poisson structures with isolated singularities over the plane. It reveals that there exists a GAGA-type phenomenon. We give an explicit description via generators and relations for the case of simple singularities.

Quadratic algebras arising from Hopf operads generated by a single element

The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated by a single element of arbitrary arity. We explain why the dual space to the space of $n$-ary operations in this operads are quadratic and Koszul algebras. We give the detailed description of generators, relations and a certain monomial basis in these algebras.