Quadratic relations of the deformed $W$-superalgebra ${\cal W}_{q, t}(\mathfrak{sl}(2|1))$

TL;DR

This paper extends the understanding of deformed $W$-superalgebras by introducing higher currents and establishing quadratic relations, providing a diagram-independent algebraic presentation.

Contribution

It introduces higher $W$-currents and derives quadratic relations that are independent of Dynkin diagram choices, enabling a generator-relation definition of the algebra.

Findings

Higher $W$-currents are constructed.

Quadratic relations among currents are established.

Relations are independent of Dynkin diagram choices.

Abstract

We revisit the free field construction of the deformed $W$-superalgebras ${\cal W}_{q,t}(\mathfrak{sl}(2|1))$ by J. Ding and B. Feigin, {\it Contemp.Math.}{\bf 248}, 83-108 (1998), where the basic $W$-current and screening currents have been found. In this paper we introduce higher $W$-currents and obtain a closed set of quadratic relations among them. These relations are independent of the choice of Dynkin diagrams for the superalgebra $\mathfrak{sl}(2|1)$, though the screening currents are not. This allows us to define ${\cal W}_{q,t}(\mathfrak{sl}(2|1))$ by generators and relations.