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Odd nilpotent element and $\mathfrak{osp}(1|2)$-subalgebra in $\mathfrak{gl}(m|n)$

Junseo Ko

TL;DR

The paper extends the classical Jacobson–Morozov framework to the Lie superalgebra $\mathfrak{gl}(m|n)$ by studying when an odd nilpotent element lies in an $\mathfrak{osp}(1|2)$-subalgebra. It introduces super Jordan matrices and a combinatorial orbit parameterization to characterize exactly which odd nilpotents can be embedded into an $\mathfrak{osp}(1|2)$-subalgebra via a $G_{\bar{0}}$-orbit. The main theorem asserts that such an embedding exists if and only if the element lies in the $G_{\bar{0}}$-orbit of a super Jordan matrix whose blocks are all of odd size. The authors provide explicit constructions of the $\mathfrak{osp}(1|2)$-subalgebra and give a detailed, blockwise proof showing the necessary and sufficient conditions on the block structure. This work clarifies the super-analogue of the Jacobson–Morozov correspondence and yields concrete criteria for embedding $\mathrm{OSp}(1|2)$ into $\mathrm{GL}(m|n)$ with a prescribed odd nilpotent element.

Abstract

In this paper, we investigate the conditions under which an odd nilpotent element in $\mathfrak{gl}(m|n)$ lies inside an $\mathfrak{osp}(1|2)$-subalgebra. In the case of the classical Lie algebra $\mathfrak{gl}_m$, every nilpotent element can be embedded into an $\mathfrak{sl}_2$-subalgebra, which is the result of the Jacobson-Morozov Theorem. In the case of the Lie superalgebra $\mathfrak{gl}(m|n)$, we define super Jordan matrices and prove that an odd nilpotent element $e$ is contained in an $\mathfrak{osp}(1|2)$-subalgebra if and only if $e$ lies in the orbit of a super Jordan matrix consisting only of super Jordan blocks of odd size.

Odd nilpotent element and $\mathfrak{osp}(1|2)$-subalgebra in $\mathfrak{gl}(m|n)$

TL;DR

The paper extends the classical Jacobson–Morozov framework to the Lie superalgebra by studying when an odd nilpotent element lies in an -subalgebra. It introduces super Jordan matrices and a combinatorial orbit parameterization to characterize exactly which odd nilpotents can be embedded into an -subalgebra via a -orbit. The main theorem asserts that such an embedding exists if and only if the element lies in the -orbit of a super Jordan matrix whose blocks are all of odd size. The authors provide explicit constructions of the -subalgebra and give a detailed, blockwise proof showing the necessary and sufficient conditions on the block structure. This work clarifies the super-analogue of the Jacobson–Morozov correspondence and yields concrete criteria for embedding into with a prescribed odd nilpotent element.

Abstract

In this paper, we investigate the conditions under which an odd nilpotent element in lies inside an -subalgebra. In the case of the classical Lie algebra , every nilpotent element can be embedded into an -subalgebra, which is the result of the Jacobson-Morozov Theorem. In the case of the Lie superalgebra , we define super Jordan matrices and prove that an odd nilpotent element is contained in an -subalgebra if and only if lies in the orbit of a super Jordan matrix consisting only of super Jordan blocks of odd size.
Paper Structure (10 sections, 14 theorems, 101 equations)

This paper contains 10 sections, 14 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Lie superalgebra
  4. Nilpotent orbits in semisimple Lie algebra
  5. Odd nilpotent orbits in $\mathfrak{gl}(m|n)$
  6. Matrix realization of odd nilpotent orbit representatives in $\mathfrak{gl}(m|n)$
  7. Construction of $\mathfrak{sl}_2$-triple of $E$
  8. Jacobson-Morozov Theorem analogue in $\mathfrak{gl}(m|n)$
  9. Construction of $\mathfrak{osp}(1|2)$-subalgebra
  10. Proof of the theorem

Key Result

Theorem 1.1

Let $\mathfrak{g} = \mathfrak{gl}(m|n)$, and $\text{Lie}(G_{\bar{0}}) = \mathfrak{g}_{\bar{0}}$ for some semisimple, simply connected group $G_{\bar{0}}$. Then an odd nilpotent element $e \in \mathfrak{g}$ is contained in an $\mathfrak{osp}(1|2)$-subalgebra of $\mathfrak{g}$ if and only if $e$ lies

Theorems & Definitions (31)

  • Theorem 1.1: Theorem \ref{['thm;theorem1']}
  • Definition
  • Definition
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition
  • Remark
  • Theorem 2.4: Jacobson-Morozov
  • Theorem 2.5: Kostant
  • ...and 21 more