Braid group action and quantum affine superalgebra for type $\mathfrak{osp}(2m+1|2n)$

TL;DR

This paper explores the structure of quantum affine superalgebras related to the orthosymplectic Lie superalgebra, establishing an isomorphism between different algebraic presentations using braid group actions.

Contribution

It introduces a braid group action to define quantum root vectors and provides an efficient method to verify isomorphisms between algebraic presentations.

Findings

Defined the Drinfeld presentation for the quantum affine superalgebra

Established an isomorphism between Drinfeld-Jimbo and Drinfeld presentations

Presented a method for verifying algebraic isomorphisms efficiently

Abstract

In this paper, we investigate the structure of the quantum affine superalgebra associated with the orthosymplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$ for $m\geqslant 1$. The Drinfeld-Jimbo presentation for this algebra, denoted as $U_q[\mathfrak{osp}(2m+1|2n)^{(1)}]$, was originally introduced by H. Yamane. We provide the definition of the Drinfeld presentation $\mathcal{U}_q[\mathfrak{osp}(2m+1|2n)^{(1)}]$. To establish the isomorphism between the Drinfeld-Jimbo presentation and the Drinfeld presentation of the quantum affine superalgebra for type $\mathfrak{osp}(2m+1|2n)$, we introduce a braid group action to define quantum root vectors of the quantum superalgebra. Specifically, we present an efficient method for verifying the isomorphism between two presentations of the quantum affine superalgebra associated with the type $\mathfrak{osp}(2m+1|2n)$.