一类量子Koszul 代数的Hochschild 上同调

本文利用组合的方法, 详细地计算了一类量子Koszul 代数Λ_q (q ∈ k \{0}) 的各阶Hochschild 上同调空间的维数, 清晰地刻划了代数Λ_q 的Hochschild 上同调的cup 积, 确定了代数Λ_q 的Hochschild上同调环HH^*(Λ_q) 模去幂零元生成的理想N 的结构, 证明了当q 为单位根时, HH^*(Λ_q)/N 作为代数不是有限生成的, 从而为Snashall-Solberg 猜想(即HH^*(Λ)/N 作为代数是有限生成的) 提供了更多反例.

Hochschild cohomology of a class of quantized Koszul algebras

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