Interrelations between Quantum Groups and Reflection Equation (Braided) Algebras

TL;DR

This paper explores the deep algebraic relationships between quantum groups and braided algebras, revealing their covariant structures and geometric implications within the framework of differential complexes and Hopf algebras.

Contribution

It establishes the covariant comodule structures of differential complexes over braided matrix algebras and their relations to quantum group extensions, highlighting new geometric insights.

Findings

Differential complex over braided matrix algebra is a covariant comodule.

Algebra of quantum groups acts as a covariant braided comodule.

Results reveal geometric aspects of quantum group and braided algebra interrelations.

Abstract

We show that the differential complex $\Omega_{B}$ over the braided matrix algebra $BM_{q}(N)$ represents a covariant comodule with respect to the coaction of the Hopf algebra $\Omega_{A}$ which is a differential extension of $GL_{q}(N)$. On the other hand, the algebra $\Omega_{A}$ is a covariant braided comodule with respect to the coaction of the braided Hopf algebra $\Omega_{B}$. Geometrical aspects of these results are discussed.