Cris Negron

We consider the finite generation property for cohomology of a finite tensor category C, which requires that the self-extension algebra of the unit Ext*_C(1,1) is a finitely generated algebra and that, for each object V in C, the graded extension group Ext*_C(1,V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C. For example, the stated result holds when C is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0, we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes.

For a simple Lie algebra g \mathsf {g} of type A A , D D , E E we show that any Belavin-Drinfeld triple on the Dynkin diagram of g \mathsf {g} produces a collection of Drinfeld twists for Lusztig’s small quantum group u q ( g ) u_q(\mathsf {g}) . These twists give rise to new finite-dimensional factorizable Hopf algebras, i.e., new small quantum groups. For any Hopf algebra constructed in this manner, we identify the group of grouplike elements, identify the Drinfeld element, and describe the irreducible representations of the dual in terms of the representation theory of the parabolic subalgebra(s) in g \mathsf {g} associated to the given Belavin-Drinfeld triple. We also produce Drinfeld twists of u q ( g ) u_q(\mathsf {g}) which express a known algebraic group action on its category of representations and pose a subsequent question regarding the classification of all twists.

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne--Mumford stacks in general. We discuss relationships with orbifold cohomology, extending Ruan's cohomological conjectures. This employs a trivialization of the determinants in the case of a symplectic group action on a symplectic variety X, which requires (for the cup product) a nontrivial normalization missing in previous literature.

Consider a field k of characteristic p > 0, the r-th Frobenius kernel ‫އ‬ (r) of a smooth algebraic group ‫,އ‬ the Drinfeld double D‫އ‬ (r) of ‫އ‬ (r) , and a finite dimensional D‫އ‬ (r)-module M. We prove that the cohomology algebra H * (D‫އ‬ (r) , k) is finitely generated and that H * (D‫އ‬ (r) , M) is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras θ r : H * ‫އ(‬ (r) , k) ⊗ S(g) → H * (D‫އ‬ (r) , k), which offers an approach to support varieties for D‫އ‬ (r)-modules. For many examples of interest, θ r is injective and induces an isomorphism of associated reduced schemes. For M an irreducible D‫އ‬ (r)-module, θ r enables us to identify the support variety of M in terms of the support variety of M viewed as a ‫އ‬ (r)-module.

Consider a field k of characteristic p > 0, G_r the r-th Frobenius kernel of a smooth algebraic group G, DG_r the Drinfeld double of G_r, and M a finite dimensional DG_r-module. We prove that the cohomology algebra H*(DG_r,k) is finitely generated and that H*(DG_r,M) is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras \theta_r:H*(G_r,k) \otimes S(g) \to H*(DG_r,k) which offers an approach to support varieties for DG_r-modules. For many examples of interest, \theta_r is injective and induces an isomorphism of associated reduced schemes. Additionally, for M an irreducible DG_r-module, \theta_r enables us to identify the support variety of M in terms of the support variety of M viewed as a G_r-module.

We classify Drinfeld twists for the quantum Borel subalgebra u_q(b) in the Frobenius-Lusztig kernel u_q(g), where g is a simple Lie algebra over C and q an odd root of unity. More specifically, we show that alternating forms on the character group of the group of grouplikes for u_q(b) generate all twists for u_q(b), under a certain algebraic group action. This implies a simple classification of Hopf algebras whose categories of representations are tensor equivalent to that of u_q(b). We also show that cocycle twists for the corresponding De Concini-Kac algebra are in bijection with alternating forms on the aforementioned character group.

We apply new techniques to Gerstenhaber brackets on the Hochschild cohomology of a skew group algebra formed from a polynomial ring and a finite group (in characteristic 0). We show that the Gerstenhaber brackets can always be expressed in terms of Schouten brackets. We obtain as consequences some conditions under which brackets are always 0, strengthening known results.

For a simple Lie algebra L of type A, D, E we show that any Belavin-Drinfeld triple on the Dynkin diagram of L produces a collection of Drinfeld twists for Lusztig's small quantum group u_q(L). These twists give rise to new finite-dimensional factorizable Hopf algebras, i.e. new small quantum groups. For any Hopf algebra constructed in this manner, we identify the group of grouplike elements, identify the Drinfeld element, and describe the irreducible representations of the dual in terms of the representation theory of the parabolic subalgebra(s) in L associated to the given Belavin-Drinfeld triple. We also produce Drinfeld twists of u_q(L) which express a known algebraic group action on its category of representations, and pose a subsequent question regarding the classification of all twists.