Gerstenhaber brackets on Hochschild cohomology of twisted tensor products

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.…”

An alternate approach to the Lie bracket on Hochschild cohomology

“…These two well known classes of examples, in Sections 4 and 5, serve merely to illustrate our techniques here. A new class of examples is given in [10]: Brackets are computed there for the quantum complete intesections Λ q := k x, y /(x 2 , y 2 , xy + qyx) for various (nonzero) values of a parameter q in a field k. The algebra structure of Hochschild cohomology of Λ q had been computed by Buchweitz, Green, Madsen, and Solberg [3]. Grimley, Nguyen, and the second author [10] used the techniques of the current paper to compute Gerstenhaber brackets directly on the Koszul resolution of Λ q .…”

“…A new class of examples is given in [10]: Brackets are computed there for the quantum complete intesections Λ q := k x, y /(x 2 , y 2 , xy + qyx) for various (nonzero) values of a parameter q in a field k. The algebra structure of Hochschild cohomology of Λ q had been computed by Buchweitz, Green, Madsen, and Solberg [3]. Grimley, Nguyen, and the second author [10] used the techniques of the current paper to compute Gerstenhaber brackets directly on the Koszul resolution of Λ q . They did not need to know explicit formulas for chain maps between the bar and Koszul resolutions, as these were not used; it suffices to know existence of such maps satisfying some conditions.…”

The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings

“…the support variety theory of [30] and the applications to loop space (co)homology [22,5]), we are also interested in it for its value as a derived invariant and for the assisting role it plays in computations of the graded Lie structure on Hochschild cohomology (see e.g. [8]). 1 We also believe that the relationship established in Theorem 1.1 is somewhat fundamental.…”