PBW-deformations of smash products involving Hopf algebra of Kac-Paljutkin type

Abstract

Let $H_{2n^2}$ be the Kac-Paljutkin type Hopf algebra of dimension $2n^2$, $A$ its graded Koszul Artin-Schelter regular $H_{2n^2}$-module algebra of dimension $2$, $A^{!}$ the Koszul dual of $A$, and $A_c^{\mathrm{op}}$ the braided-opposite algebra of $A$. This paper describes $(0,1)$-degree PBW-deformations of the smash product $A♯H_{2n^2}$ and those of $A^{!}♯H_{2n^2}$ under the condition that the Koszul dual $A^{!}$ of $A$ is also an $H_{2n^2}$-module algebra. Also, $0$-degree PBW-deformations of $(A{\otimes }^c{A}_c^{\mathrm{op}})♯H_{2n^2}$ are explored, where $A{\otimes }^c{A}_c^{\mathrm{op}}$ is the associated braided tensor product algebra.

30 pages