# Factorizable Module Algebras

## TL;DR

This paper introduces and studies a broad class of factorizable $rak{g}$-module algebras, generalizing Gauss factorization, with quantum analogs that are acted upon by dual quantum groups, expanding the understanding of algebraic structures related to Lie groups.

## Contribution

It defines and explores factorizable module algebras, including their quantum versions and dual actions, broadening the framework for algebraic and quantum group structures.

## Key findings

- Factorizable algebras include coordinate algebras of reductive groups and related structures.
- Tensor products of factorizable algebras are also factorizable.
- Quantum factorizable algebras are acted on by the dual quantum group $U_q(rak{g}^*)$.

## Abstract

The aim of this paper is to introduce and study a large class of $\mathfrak{g}$-module algebras which we call factorizable by generalizing the Gauss factorization of (square or rectangular) matrices. This class includes coordinate algebras of corresponding reductive groups $G$, their parabolic subgroups, basic affine spaces and many others. It turns out that tensor products of factorizable algebras are also factorizable and it is easy to create a factorizable algebra out of virtually any $\mathfrak{g}$-module algebra. We also have quantum versions of all these constructions in the category of $U_q(\mathfrak{g})$-module algebras. Quite surprisingly, our quantum factorizable algebras are naturally acted on by the quantized enveloping algebra $U_q(\mathfrak{g}^*)$ of the dual Lie bialgebra $\mathfrak{g}^*$ of $\mathfrak{g}$.