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Lie algebras generated by reflections in types BCD

Christopher M. Drupieski, Jonathan R. Kujawa

Abstract

We consider the group algebra over the field of complex numbers of the Weyl group of type B (the hyperoctahedral group, or the group of signed permutations) and of the Weyl group of type D (the demihyperoctahedral group, or the group of even-signed permutations), viewed as Lie algebras via the commutator bracket, and determine the structure of the Lie subalgebras generated by the sets of reflections.

Lie algebras generated by reflections in types BCD

Abstract

We consider the group algebra over the field of complex numbers of the Weyl group of type B (the hyperoctahedral group, or the group of signed permutations) and of the Weyl group of type D (the demihyperoctahedral group, or the group of even-signed permutations), viewed as Lie algebras via the commutator bracket, and determine the structure of the Lie subalgebras generated by the sets of reflections.

Paper Structure

This paper contains 56 sections, 49 theorems, 158 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Lie algebra of transpositions
  3. The Lie algebra of reflections in types BCD
  4. Further questions
  5. Conventions
  6. Acknowledgements
  7. Type B classical representation theory
  8. Type B Weyl groups
  9. Simple modules for Type B Weyl groups
  10. Weight space decompositions of Type B simple modules
  11. Orthogonal forms
  12. Restriction of Type B simple modules
  13. Type D classical representation theory
  14. Type D Weyl groups
  15. Simple modules for Type D Weyl groups
  16. ...and 41 more sections

Key Result

Proposition 1

For $n \geq 2$, the Artin--Wedderburn maps induce Lie algebra homomorphisms and where $\mathcal{L}_{n}$ is isomorphic to the Lie algebra $\mathfrak{s}_{n}'$ determined by Marin, and where the notation $\star$ means that the term $\mathfrak{d}_{\left\{ \lambda,- \right\}}^\star$ is omitted if $n/2$ is odd.

Theorems & Definitions (88)

  • Proposition : \ref{['prop:restriction-of-AW-map']}
  • Theorem : \ref{['thm:main-theorem']}
  • Lemma 2.2.1
  • proof
  • Theorem 2.2.2
  • Theorem 2.4.1
  • Remark 2.4.2
  • Lemma 2.4.3
  • proof
  • Lemma 2.4.4
  • ...and 78 more