Presentations for small reflection equation algebras of type A

TL;DR

This paper provides explicit generator-and-relation presentations for small reflection equation algebras of type A, clarifying their structure as quotients of infinite-dimensional algebras with added relations.

Contribution

It introduces new presentations for these algebras, linking finite-dimensional quantum groups to their infinite-dimensional counterparts through specific relations.

Findings

Presentations as quotients of infinite-dimensional algebras.

Relations encode twisting of nilpotency and unipotency.

Valid for integral forms of the algebras.

Abstract

We give presentations, in terms of the generators and relations, for the reflection equation algebras of type $GL_n$ and $SL_n$, i.e., the covariantized algebras of the dual Hopf algebras of the small quantum groups of $\mathfrak{gl}_n$ and $\mathfrak{sl}_n$. Our presentations display these algebras as quotients of the infinite-dimensional reflection equation algebras of types $GL_n$ and $SL_n$ by identifying additional relations that correspond to twisting the nilpotency and unipotency relations of the finite-dimensional quantum function algebras. The presentations are valid for appropriately defined integral forms of these algebras.