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 $A^{\mathrm{op}}_c$ the braided-opposite algebra of $A$. This paper describes $(0, 1)$-degree PBW-deformations of the smash product $A \sharp H_{2n^2}$ and those of $A^! \sharp\, H_{2n^2}$ under the condition that the Koszul dual $A^!$ of $A$ is also an $H_{2n^2}$-module algebra. Also, $0$-degree PBW-deformations of $(A \otimes^c A^{\mathrm{op}}_c) \sharp\, H_{2n^2}$ are explored, where $A \otimes^c A^{\mathrm{op}}_c$ is the associated braided tensor product algebra.