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Structure and Representation Theory of basic simple $\mathbb{Z}_2\times \mathbb{Z}_2$-graded color Lie algebras

Spyridon Afentoulidis-Almpanis

Abstract

We adapt methods from the theory of complex semisimple Lie algebras to develop a root theory for a class of simple $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded (color) Lie algebras, which we call basic. As an application, assuming that the Cartan subalgebra is self-centralizing, we classify all finite-dimensional representations of these algebras by proving a highest weight theorem and a complete reducibility theorem.

Structure and Representation Theory of basic simple $\mathbb{Z}_2\times \mathbb{Z}_2$-graded color Lie algebras

Abstract

We adapt methods from the theory of complex semisimple Lie algebras to develop a root theory for a class of simple -graded (color) Lie algebras, which we call basic. As an application, assuming that the Cartan subalgebra is self-centralizing, we classify all finite-dimensional representations of these algebras by proving a highest weight theorem and a complete reducibility theorem.
Paper Structure (12 sections, 13 theorems, 146 equations)

This paper contains 12 sections, 13 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Basic $\mathbb{Z}_2\times\mathbb{Z}_2$-graded Lie algebras
  3. $\mathbb{Z}_2\times \mathbb{Z}_2$-graded Lie algebras
  4. Basic simple $\mathbb{Z}_2\times \mathbb{Z}_2$-graded Lie algebras
  5. Root theory for basic simple $\mathbb{Z}_2\times\mathbb{Z}_2$-graded Lie algebras
  6. Applications to finite-dimensional representations
  7. Highest Weight Theorem
  8. Theorem of complete reducibility
  9. Two examples and a question
  10. The case of $\mathfrak{so}(4,2,2,2)_\mathbb{C}$
  11. The case of $\mathfrak{so}(4,2,1,1)_{\mathbb{C}}$
  12. A question

Key Result

Lemma 2.5

If $\mathfrak{g}$ is a simple $\mathbb{Z}_2\times\mathbb{Z}_2$-graded Lie algebra with $\mathfrak{g}^{(0,0)}$ being reductive and nonabelian, then the Killing form $K$ of $\mathfrak{g}$ is nondegenerate so that $\mathfrak{g}$ is basic.

Theorems & Definitions (30)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 20 more