Principal SUSY and nonSUSY W-algebras and their Zhu algebras

TL;DR

This paper proves the supersymmetry of certain W-algebras associated with Lie superalgebras and establishes their isomorphism with Zhu algebras, advancing understanding of their algebraic structures and symmetries.

Contribution

It demonstrates the isomorphism between principal SUSY and nonSUSY W-algebras and their Zhu algebras, revealing new structural insights.

Findings

W-algebra $W^k(rak{g},F)$ is isomorphic to SUSY W-algebra via screening operators

Finite SUSY W-algebras are isomorphic to Zhu algebras of SUSY W-algebras

Finite SUSY principal W-algebra is isomorphic to finite principal W-algebra

Abstract

This paper consists of two parts. In the first part, we prove that when $\mathfrak{g}$ is a simple basic Lie superalgebra with a principal odd nilpotent element $f$, the W-algebra $W^k(\mathfrak{g}, F)$ for $F=-\frac{1}{2}[f,f]$ is isomorphic to the SUSY W-algebra $W^k(\bar{\mathfrak{g}},f)$ via screening operators, which implies the supersymmetry of $W^k(\mathfrak{g}, F)$. In the second part, we show that a finite SUSY W-algebra, which is a Hamiltonian reduction of $U(\widetilde{\mathfrak{g}})$ for the SUSY Takiff algebra $\widetilde{\mathfrak{g}}=\mathfrak{g}\otimes \wedge(\theta)$ is isomorphic to the Zhu algebra of a SUSY W-algebra. As a corollary, we show that a finite SUSY principal W-algebra is isomorphic to a finite principal W-algebra.