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

Wan Keng Cheong, Ngau Lam

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.

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

Abstract

Let be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra , where is , or . We show, using super duality, that the singular eigenvectors of the (super) Gaudin Hamiltonians for on 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 are diagonalizable on the space spanned by singular vectors of and hence on . In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra and the symplectic Lie algebra , on the tensor product of infinite-dimensional unitarizable irreducible highest weight modules.
Paper Structure (14 sections, 29 theorems, 73 equations)

This paper contains 14 sections, 29 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General linear superalgebra $\widetilde{\mathcal{G}}^{\mathfrak{a}}$
  4. Ortho-symplectic superalgebra $\widetilde{\mathcal{G}}^{\mathfrak{c}}$ and its subalgebras
  5. Ortho-symplectic superalgebra $\widetilde{\mathcal{G}}^{\mathfrak{d}}$ and its subalgebras
  6. Dynkin diagrams
  7. Central extensions
  8. Parabolic BGG categories and super duality
  9. Unitarizable $\overline{\mathcal{G}}^{\mathfrak x}[m]_n$-modules
  10. $*$-structures on $\overline{\mathcal{G}}^{\mathfrak x}[m]_n$ and $\overline{\mathfrak g}^{\mathfrak x}[m]_n$
  11. Unitarizable $\overline{\mathcal{G}}^{\mathfrak a}[m]_n$-modules
  12. Unitarizable modules over $\overline{\mathcal{G}}^{\mathfrak c}[m]_n$ and $\overline{\mathcal{G}}^{\mathfrak d}[m]_n$
  13. Gaudin Hamiltonians on modules over $\widetilde{{\mathfrak g}}$, ${\mathfrak g}[m]_n$ and $\overline{\mathfrak g}[m]_n$
  14. Gaudin Hamiltonians on modules over $\overline{\mathcal{G}}[m]_n$

Key Result

Theorem 1.1

There exist one-to-one correspondences relating the sets of singular eigenvectors of the Gaudin Hamiltonians for $\tilde{\mathfrak g}$, ${\mathfrak g}[m]$ and $\overline{\mathfrak g}[m]$ for $m \in {\mathbb Z}_+$.

Theorems & Definitions (51)

  • Theorem 1.1: Theorem \ref{['eigenvector-C1']}
  • Theorem 1.2: Theorem \ref{['dom-uni-1']} and Theorem \ref{['dom-uni-2']}
  • Remark 1.3
  • Remark 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • ...and 41 more