On dynamical smash product

TL;DR

This paper explores the structure of the dynamical smash product in the context of the dynamical Yang-Baxter equation, generalizing the smash product concept and analyzing modules induced from one-dimensional representations.

Contribution

It introduces and studies the dynamical smash product associated with Hopf algebras and base algebras, extending the classical smash product framework.

Findings

Construction of an analog of the Galois map for the dynamical smash product

Analysis of modules induced from one-dimensional representations of the base algebra

Generalization of the smash product in the setting of dynamical Yang-Baxter theory

Abstract

In the theory of dynamical Yang-Baxter equation, with any Hopf algebra $H$ and a certain $H$-module and $H$-comodule algebra $L$ (base algebra) one associates a monoidal category. Given an algebra $A$ in that category, one can construct an associative algebra $A\rtimes L$, which is a generalization of the ordinary smash product when $A$ is an ordinary $H$-algebra. We study this "dynamical smash product" and its modules induced from one-dimensional representation of the subalgebra $L$. In particular, we construct an analog of the Galois map $A\otimes_{A^H} A\to A\otimes H^*$.