Cup product and Gerstenhaber bracket on Hochschild cohomology of a family of quiver algebras

Abstract

We present a cup product formula on the Hochschild cohomology of a family of quiver algebras. We use the formula to determine the set of homogeneous non-nilpotent Hochschild cocycles and construct a canonical isomorphism between Hochschild cohomology modulo nilpotents and a subalgebra of $k[x,y]$ that is not finitely generated. For some members of the family, we present the Gerstenhaber ideal of homogeneous nilpotent cocycles using homotopy lifting technique. We then determine their Hochschild cohomology modulo the weak Gerstenhaber ideal generated by nilpotent elements, thereby providing an answer to a question of Reiner Hermann.

Keywords:

• Cup products
• Gerstenhaber brackets
• Hochschild cohomology
• Koszul algebras
• Snashall-Solberg finite generation conjecture

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16E40
• Secondary: 16S37

Acknowledgment

The author thanks his thesis advisor Dr. Sarah Witherspoon for useful discussions, reading through the manuscript and making many useful suggestions.

Additional information

Funding

Author was partially supported by NSF grants 1665286 and 2001163 during Summer 2020 and Spring 2021 while completing his graduate studies at Texas A&M University.