On the structure of $End_{u_k(2)}(\Omega_k^{\otimes r})$

TL;DR

This paper investigates the algebraic structure of the endomorphism algebra of tensor powers of the natural module for the infinitesimal quantum group $u_k(2)$ over a field containing a primitive root of unity, providing a detailed algebraic characterization.

Contribution

It determines the basic algebra of the endomorphism algebra of tensor powers of the natural module for $u_k(2)$, advancing understanding of its representation theory.

Findings

Identification of the basic algebra structure

Explicit description of endomorphism algebra

Insights into the module category of $u_k(2)$

Abstract

Let $u_k(2)$ be the infinitesimal quantum $\frak{gl}_2$ over $k$, where $k$ is a field containing an $l$th primitive root $\epsilon$ of 1 with $l\geq 3$ {\it odd}. We will determine the basic algebra for ${u_k(2)}(\Omega_k^{\otimes r})$, where $\Omega_k$ is the natural module for $u_k(2)$.