PBW-deformations of smash products involving Hopf algebra of Kac-Paljutkin type
Yujie Gao, Shilin Yang
TL;DR
The paper addresses PBW-deformations of smash products involving the Kac-Paljutkin type Hopf algebra $H_{2n^2}$ acting on dimension-2 AS-regular algebras, their Koszul duals, and braided products. It provides a constructive classification of nontrivial $(0,1)$-degree PBW-deformations for $A\sharp H_{2n^2}$ and $A^!\sharp H_{2n^2}$, as well as 0-degree deformations for braided products, by exploiting the PBW framework via a bilinear deformation map $\kappa$ and imposing invariance/compatibility constraints. The results include explicit conditions on the indexing parameters $(k,l)$ and $n$, and detailed formulas for $\kappa$ in various cases, thereby extending PBW deformation knowledge from $H_8$ to the broader family $H_{2n^2}$ and including Koszul duals and braided constructions. This work advances understanding of deformations for Hopf-action algebras and braided products, with potential implications for noncommutative algebraic structures with quantum group symmetry.
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.