On Kostant's theorem for the Lie superalgebra Q(n)

TL;DR

This paper investigates finite W-algebras for the Lie superalgebra Q(n), proving they satisfy the Amitsur-Levitzki identity and providing explicit descriptions and realizations as quotients of super-Yangians.

Contribution

It establishes the Amitsur-Levitzki identity for these W-algebras and explicitly describes the algebra for Q(n) as a quotient of the super-Yangian Q(1).

Findings

Finite W-algebras for Q(n) satisfy the Amitsur-Levitzki identity.

All irreducible representations of these algebras are finite-dimensional.

Explicit generators and relations for the W-algebra of Q(n) are provided.

Abstract

In this paper we study finite W-algebras for basic classical superalgebras and Q(n) associated to the regular even nilpotent coadjoint orbits. We prove that this algebra satisfies the Amitsur-Levitzki identity and therefore all its irreducible representations are finite-dimensional. In the case of Q(n) we give an explicit description of the W-algebra in terms of generators and relation and realize it as a quotient of the super-Yangian of Q(1).