Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Narrow, wide, and $λ$-wide regular subalgebras of semisimple Lie algebras

Andrew Douglas, Joe Repka

Abstract

A subalgebra of a semisimple Lie algebra is wide if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. From a finer viewpoint, a subalgebra is $λ$-wide if the simple module of a semisimple Lie algebra of highest weight $λ$ remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restriction of all non-trivial simple modules to the subalgebra have proper decompositions. We determine necessary and sufficient conditions for regular subalgebras of semisimple Lie algebras to be $λ$-wide. As a natural consequence, we establish necessary and sufficient conditions for regular subalgebras to be wide, a result which has already been established by Panyushev for essentially all regular solvable subalgebras. Next, we show that establishing whether or not a regular subalgebra of a simple Lie algebra is wide does not require consideration of all simple modules. It is necessary and sufficient to only consider the adjoint representation. Finally, we show that a regular subalgebra of the special linear algebra $\mathfrak{sl}_{n+1}$ is either narrow or wide; this property does not hold for non-regular subalgebras of $\mathfrak{sl}_{n+1}$.

Narrow, wide, and $λ$-wide regular subalgebras of semisimple Lie algebras

Abstract

A subalgebra of a semisimple Lie algebra is wide if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. From a finer viewpoint, a subalgebra is -wide if the simple module of a semisimple Lie algebra of highest weight remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restriction of all non-trivial simple modules to the subalgebra have proper decompositions. We determine necessary and sufficient conditions for regular subalgebras of semisimple Lie algebras to be -wide. As a natural consequence, we establish necessary and sufficient conditions for regular subalgebras to be wide, a result which has already been established by Panyushev for essentially all regular solvable subalgebras. Next, we show that establishing whether or not a regular subalgebra of a simple Lie algebra is wide does not require consideration of all simple modules. It is necessary and sufficient to only consider the adjoint representation. Finally, we show that a regular subalgebra of the special linear algebra is either narrow or wide; this property does not hold for non-regular subalgebras of .
Paper Structure (6 sections, 17 theorems, 18 equations)

This paper contains 6 sections, 17 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Semisimple Lie algebras and their modules
  4. Closed subsets of root systems and regular subalgebras
  5. Wide and $\lambda$-wide regular subalgebras
  6. Regular subalgebras of $\mathfrak{sl}_{n+1}$

Key Result

Lemma 2.1

sopkina$T^r$ is a closed root subsystem of $\Phi$. For any two roots $\alpha \in T^u$ and $\beta \in T$ such that $\alpha+\beta$ is a root, we have that $\alpha+\beta \in T^u$. In particular, $T^u$ is closed.

Theorems & Definitions (28)

  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 18 more