Classification of Hopf superalgebras associated with quantum special linear superalgebra at roots of unity using Weyl groupoid

TL;DR

This paper classifies Hopf superalgebras related to quantum special linear superalgebras at roots of unity using Weyl groupoids, constructing explicit bases, twists, and universal R-matrices, and analyzing their classical limits.

Contribution

It introduces a supercategory approach to define Weyl groupoids, classifies quantum superalgebras via twists and isomorphisms, and constructs explicit bases and R-matrices for these structures.

Findings

Quantum superalgebras are classified via Weyl groupoids.

Explicit PBW bases and universal R-matrices are constructed.

Connections between quantum superalgebras and classical Lie superalgebras are established.

Abstract

We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the quantum Weyl groupoid. The Weyl groupoid of $\mathfrak{sl}(m|n)$ is constructed to this end as some supercategory. We prove that in this case quantum superalgebras associated with Dynkin diagrams are isomorphic as superalgebras. It is shown how these quantum superalgebras considered as Hopf superalgebras are connected via twists and isomorphisms. We explicitly construct these twists using the Lusztig isomorphisms considered as elements of the Weyl quantum groupoid. We build a PBW basis for each quantum superalgebra, and investigate how quantum superalgebras are connected with their classical limits, i. e. Lie superbialgebras. We find explicit multiplicative formulas for universal $R$-matrices, describe relations between them for each realization and classify Hopf superalgebras and triangular structures for the quantum superalgebra $U_q(\mathfrak{sl}(m|n))$.