Hochschild cohomology of group extensions of quantum complete intersections

Abstract

We formulate the Gerstenhaber algebra structure of Hochschild cohomology of finite group extensions of some quantum complete intersections. When the group is trivial, this work characterizes the graded Lie brackets on Hochschild cohomology of these quantum complete intersections, previously only known for a few cases. As an example, we compute the algebra structure for two generator quantum complete intersections extended by select groups.

A Geometric Approach to Hochschild Cohomology of the Exterior Algebra

“…Also [α, γ] ∈ HH n−1 odd ( V ), which is trivial unless n = 1. This statement in particular recovers the results regarding the additive and cup product structure in [8], [5], [30], [36] and [27], as well as the results about the Gerstenhaber structure in [19] and [18]. Remark 5.6.…”

“…As for what is known, computations of the Hochschild (co)homology groups of the exterior algebra and the cohomological ring structure have been given in various contexts : as a special case of the results for quantum complete intersections ( [8], [5], [30]), from direct combinatorial techniques [36], and from algebraic Morse theory [27]. Recently, the Gerstenhaber bracket has been described for quantum complete intersections and their group extensions ( [19], [18]), building off the work of [6] on twisted tensor products and of [29] on defining the bracket on complexes other than the bar complex. Furthermore, the general theory of [26] and [34] on Frobenius algebras with semisimple Nakayama automorphism ensures that Hochschild cohomology of the exterior algebra is a Batalin-Vilkovisky (BV) algebra.…”

Truncated Quantum Drinfeld Hecke Algebras and Hochschild Cohomology

“…Thus we would like to construct a small resolution of Λ q (V ) to compute Hochschild cohomology. One such small resolution of Λ q (V ) was given in [6] which we describe in the next subsection.…”

“…With this result, we further develop the connection between Hochschild cohomology and truncated quantum Drinfeld Hecke algebras, allowing for another perspective for computations. Finally, in addition to the example mentioned previously, Section 5 contains examples of truncated quantum Drinfeld Hecke algebras arising from diagonal group actions, whose Hochschild cohomology was computed separately by the first author in [6].…”

Homotopy liftings and Hochschild cohomology of some twisted tensor products

“…These results illustrate the theoretical usefulness of homotopy liftings. In some settings they are also computationally useful: see, for example, [4,5] for some quantum complete intersections and smash products with groups, [6] for the Jordan plane, and [10] for polynomial rings and some types of cyclic group algebras. In particular, in [5,6], elementary methods allow the application of the techniques in [10] to compute Gerstenhaber brackets on the Hochschild cohomology of twisted tensor products.…”