Classcal affine W-algebras associated to Lie superalgebras

TL;DR

This paper explores the construction, properties, and generalizations of classical affine W-superalgebras linked to Lie superalgebras, establishing new methods and explicit generators.

Contribution

It introduces two methods for constructing classical affine W-superalgebras, computes explicit generators, and proposes a new class called fractional W-superalgebras with Poisson vertex algebra structures.

Findings

Classical affine W-superalgebras can be constructed via Hamiltonian reductions and quasi-classical limits.

Explicit generators for W-superalgebras associated to minimal and principal nilpotent elements are found.

A new class of fractional W-superalgebras with Poisson vertex algebra structures is introduced.

Abstract

In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras) can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum affine W-superalgebras. Also, we show that a classical finite W-superalgebra can be obtained by a Zhu algebra of a classical affine W-superalgebra. Using the definition by Hamiltonian reductions, we find free generators of a classical W-superalgebra associated to a minimal nilpotent. Moreover, we compute generators of the classical W-algebra associated to $spo(2|3)$ and its principal nilpotent. In the last part of this paper, we introduce a generalization of classical affine W-superalgebras called classical affine fractional W-superalgebras. We show these have Poisson vertex algebra structures and find generators of a fractional W-superalgebra associated to a minimal nilpotent.