Classification of irreducible unitary modules over $\mathfrak{u}(p,q|n)$

TL;DR

This paper provides a complete classification of all irreducible highest- and lowest-weight unitary modules over the Lie superalgebra (p,q|n), using duality and invariants.

Contribution

It introduces explicit necessary and sufficient conditions for classifying irreducible unitary modules over (p,q|n), combining Howe duality and quadratic invariants.

Findings

Classified all irreducible highest-weight unitary modules over (p,q|n).

Classified all irreducible lowest-weight unitary modules over (p,q|n).

Classified all irreducible unitary modules over (n|q,p) via isomorphism.

Abstract

We classify all irreducible highest-weight unitary modules over the non-compact real form $\mathfrak{u}(p,q|n)$ of the general linear Lie superalgebra $\mathfrak{gl}_{p+q|n}$. The classification is given by explicit necessary and sufficient conditions on the highest weights, and our approach combines the Howe duality for $\mathfrak{gl}_{p+q|n}$ with a quadratic invariant of the maximal compact subalgebra. Using this classification result, we also classify all irreducible lowest-weight unitary modules over $\mathfrak{u}(p,q|n)$ via duality, and all irreducible unitary modules over $\mathfrak{u}(n|q,p)$ via an isomorphism of Lie superalgebras.