$\mathfrak{R}$-matrix for quantum superalgebra $\mathfrak{sl}(2|1)$ at roots of unity and its application to centralizer algebras

TL;DR

This paper constructs the universal R-matrix for the quantum superalgebra sl(2|1) at roots of unity and explores its application to the structure of centralizer algebras, advancing understanding of quantum superalgebra representations.

Contribution

It provides a new explicit construction of the R-matrix for sl(2|1) at roots of unity and analyzes the structure of related centralizer algebras.

Findings

Explicit formula for the universal R-matrix of sl(2|1) at roots of unity

Multiplication laws for centralizer algebras in specific cases

Structural description of centralizer algebras in general case

Abstract

We consider fundamental facts from the theory of Hopf superalgebras. We use them to construct the quantum double of the quantum superalgebra $sl(2|1)$ at roots of unity. Thus we obtain a multiplicative formula for universal $R$-matrix. Next we construct an $R$-matrix to investigate parametrized family of centralizer algebras. We give multiplication laws in particular case and describe a structure of such algebras in the general case.