Shuffle algebra realization of quantum affine superalgebra $U_{v}(\hat{\mathfrak{D}}(2,1;\theta))$

TL;DR

This paper constructs a shuffle algebra realization for the positive part of the quantum affine superalgebra associated with de9(2,1;b8), extending previous results to new root systems and special cases.

Contribution

It provides a new shuffle algebra realization for the positive part of quantum affine superalgebras for any simple root system, including the d8sl(2|1) case at roots of unity.

Findings

Constructed shuffle algebra realization for d8D(2,1;b8)

Extended results to d8sl(2|1) at roots of unity

Generalized previous algebraic structures and realizations

Abstract

We give shuffle algebra realization of positive part of quantum affine superalgebra $U_{v}(\widehat{\mathfrak{D}}(2,1;\theta))$ associated to any simple root systems. We also determine the shuffle algebra associated to $\widehat{\mathfrak{sl}}(2|1)$ with odd root system when $v$ is a primitive root of unity of even order, generalizing results in \cite{FJMMT03}.