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On matrix Lie affgebras

Tomasz Brzeziński, Krzysztof Radziszewski

Abstract

Lie brackets or Lie affgebra structures on several classes of affine spaces of matrices are studied. These include general normalised affine matrices, special normalised affine matrices, anti-symmetric and anti-hermitian normalised affine matrices and special anti-hermitian normalised affine matrices. It is shown that, when retracted to the underlying vector spaces, they correspond to classical matrix Lie algebras: general and special linear, anti-symmetric, anti-hermitian and special anti-hermitian Lie algebras respectively.

On matrix Lie affgebras

Abstract

Lie brackets or Lie affgebra structures on several classes of affine spaces of matrices are studied. These include general normalised affine matrices, special normalised affine matrices, anti-symmetric and anti-hermitian normalised affine matrices and special anti-hermitian normalised affine matrices. It is shown that, when retracted to the underlying vector spaces, they correspond to classical matrix Lie algebras: general and special linear, anti-symmetric, anti-hermitian and special anti-hermitian Lie algebras respectively.
Paper Structure (3 sections, 3 theorems, 44 equations)

This paper contains 3 sections, 3 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Lie brackets on affine spaces
  3. Matrix Lie affgebras and corresponding Lie algebras

Key Result

Lemma 3.1

Let $\mathfrak{a}$ denote any of the sets $\mathrm{gna}(n,\mathbb F)$, $\mathrm{sna}(n,\mathbb F)$, $\mathrm{ona}(n)$, $\mathrm{una}(n)$ or $\mathrm{suna}(n)$. Then $\mathfrak{a}$ is a Lie affgebra with the following operations: for all $\mathbf{a}, \mathbf{b}, \mathbf{c}\in \mathfrak{a}$ and $\alpha \in \mathbb F$.

Theorems & Definitions (7)

  • Definition 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof