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Regular extreme 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. A subalgebra is narrow if the restrictions of all non-trivial simple modules to the subalgebra have proper decompositions. A semisimple Lie algebra is regular extreme if any regular subalgebra of the semisimple Lie algebra is either narrow or wide. Douglas and Repka previously showed that the simple Lie algebras of type $A_n$ are regular extreme. In this article, we show that, in fact, all simple Lie algebras are regular extreme. Finally, we show that no non-simple, semisimple Lie algebra is regular extreme.

Regular extreme 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. A subalgebra is narrow if the restrictions of all non-trivial simple modules to the subalgebra have proper decompositions. A semisimple Lie algebra is regular extreme if any regular subalgebra of the semisimple Lie algebra is either narrow or wide. Douglas and Repka previously showed that the simple Lie algebras of type are regular extreme. In this article, we show that, in fact, all simple Lie algebras are regular extreme. Finally, we show that no non-simple, semisimple Lie algebra is regular extreme.
Paper Structure (8 sections, 9 theorems, 9 equations, 2 figures)

This paper contains 8 sections, 9 theorems, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background, terminology, and notation
  3. Semisimple Lie algebras
  4. Closed subsets of root systems and regular subalgebras
  5. Simple Lie algebras and Dynkin diagrams
  6. Previous results
  7. Regular extreme simple Lie algebras
  8. Non-simple, semisimple Lie algebras

Key Result

Lemma 2.1

dr24 Let $S$ be a closed subset of $\Phi$. Suppose $\beta_1$, $\beta_2$,...,$\beta_k \in S$, and $\beta_1+\beta_2+ \cdots+\beta_k \in \Phi$. Then $\beta_1+\beta_2+\cdots +\beta_k \in S$.

Figures (2)

  • Figure 1: Dynkin diagrams of the simple Lie algebras.
  • Figure 2: A simple root path $(\beta_1, \beta_2, \beta_3, \beta_4)$ of $E_8$.

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Corollary 2.6
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 3 more