Hochschild cohomology of graded skew-gentle algebras: Gerstenhaber algebra structure and geometric interpretation
Xiuli Bian, Sibylle Schroll, Andrea Solotar, Xiao-chuang Wang, Can Wen
TL;DR
The authors compute the Hochschild cohomology ${\mathrm{HH}}^*(A)$ of graded skew-gentle algebras, including its graded-commutative cup product and Gerstenhaber bracket, using a graded CS-projective resolution and comparison maps to transfer structure from the bar resolution. They show that ${\mathrm{HH}}^*(A)$ admits explicit generators and relations, and that the deformation from graded gentle to graded skew-gentle algebras contributes a calculable finite-dimensional space $V_{\mathrm{sp}}^N$. A central achievement is a geometric interpretation: generators of ${\mathrm{HH}}^*(A)$ correspond to features of the associated graded orbifold surface (boundary components with one marked point, $G$- and $G^*$-punctures) together with the fundamental group $\pi_1(S_{\Lambda})$, with degrees controlled by combinatorial winding numbers. This ties the algebraic structure to surface topology and to partially wrapped Fukaya categories, offering a unified framework for reading Hochschild cohomology from the underlying surface data and for understanding the cup product and Gerstenhaber bracket via curves on the surface.
Abstract
In this paper we calculate the Hochschild cohomology of graded skew-gentle algebras, together with its structure as graded commutative algebra under the cup product and its Lie algebra structure given by the Gerstenhaber bracket. One of the results of this paper is that for graded skew-gentle algebras and thus also for partially wrapped Fukaya categories of orbifold surfaces with stops, their Hochschild cohomology is encoded in the underlying graded surface.