Hochschild cohomology of a class of quantized Koszul algebras

Abstract

In this paper, we mainly compute the <italic>k</italic>-dimensions of Hochschild cohomology spaces of a class of quantized Koszul algebras Λ_{<italic>q</italic>} (<italic>q</italic> ∈ k \\{0}) with the application of combinatorics, explicitly describe the cup product of Hochschild cohomology of Λ_{<italic>q</italic>}, and thus determine the structure of the Hochschild cohomology ring HH^*(Λ_{<italic>q</italic>}) of Λ_{<italic>q</italic>} modulo the ideal <italic>N</italic> generated by nilpotence. As a consequence, we show that HH^*(Λ_{<italic>q</italic>})/<italic>N</italic> is not finitely generated as an algebra when <italic>q</italic> is a root of unity, and thus provide more counterexamples for Snashall- Solberg conjecture (namely, the Hochschild cohomology ring modulo nilpotence of a finite-dimensional <italic>k</italic>-algebra is always finitely generated as algebra).