Drinfeld second realization of the quantum affine superalgebras of $D^{(1)}(2,1;x)$ via the Weyl groupoid

TL;DR

This paper derives the Drinfeld second realization for quantum affine superalgebras related to $D^{(1)}(2,1;x)$, utilizing a Weyl groupoid structure, extending Beck's methods from affine Lie algebras to superalgebras.

Contribution

It introduces a new realization of quantum affine superalgebras using Weyl groupoids, expanding the algebraic framework beyond traditional affine Weyl groups.

Findings

Established Drinfeld second realization for $D^{(1)}(2,1;x)$ superalgebras

Utilized Weyl groupoid instead of affine Weyl groups

Extended Beck's approach to superalgebra context

Abstract

We obtain Drinfeld second realization of the quantum affine superalgebras associated with the affine Lie superalgebra $D^{(1)}(2,1;x)$. Our results are analogous to those obtained by Beck for the quantum affine algebras. Beck's analysis uses heavily the (extended) affine Weyl groups of the affine Lie algebras. In our approach the structures are based on a Weyl groupoid.