Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras

TL;DR

This paper demonstrates the diagonalizability of Gaudin Hamiltonians on unitarizable modules over classical Lie superalgebras by relating their eigenvectors to those of classical Lie algebras via super duality.

Contribution

It establishes a method to diagonalize superalgebra Gaudin Hamiltonians using super duality, extending results to infinite-dimensional unitarizable modules.

Findings

Singular eigenvectors of superalgebra Gaudin Hamiltonians derived from classical Lie algebra eigenvectors.

Diagonalization of Gaudin Hamiltonians on unitarizable modules over orthogonal and symplectic Lie algebras.

Extension of diagonalization results to infinite-dimensional modules.

Abstract

Let $M$ be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra $\mathcal{G}$, where $\mathcal{G}$ is $\mathfrak{gl}(m|n)$, $\mathfrak{osp}(2m|2n)$ or $\mathfrak{spo}(2m|2n)$. We show, using super duality, that the singular eigenvectors of the (super) Gaudin Hamiltonians for $\mathcal{G}$ on $M$ can be obtained from the singular eigenvectors of the Gaudin Hamiltonians for the corresponding Lie algebras on some tensor products of finite-dimensional irreducible modules. As a consequence, the (super) Gaudin Hamiltonians for $\mathcal{G}$ are diagonalizable on the space spanned by singular vectors of $M$ and hence on $M$. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra $\mathfrak{so}(2n)$ and the symplectic Lie algebra $\mathfrak{sp}(2n)$, on the tensor product of infinite-dimensional unitarizable irreducible highest weight modules.