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Remarks on quadratic left Bol algebras

A. Nourou Issa

Abstract

In this paper the notion of a quadratic (left) Bol algebra is discussed. Several examples of quadratic Bol algebras are given and it is observed that the only two-dimensional quadratic real Bol algebras are quadratic Lie triple systems. Dual representations of Bol algebras are investigated with a particular emphasis on coadjoint representations for quadratic Bol algebras. The notion of T*-extension of a quadratic Bol algebra is introduced.

Remarks on quadratic left Bol algebras

Abstract

In this paper the notion of a quadratic (left) Bol algebra is discussed. Several examples of quadratic Bol algebras are given and it is observed that the only two-dimensional quadratic real Bol algebras are quadratic Lie triple systems. Dual representations of Bol algebras are investigated with a particular emphasis on coadjoint representations for quadratic Bol algebras. The notion of T*-extension of a quadratic Bol algebra is introduced.
Paper Structure (4 sections, 13 theorems, 29 equations)

This paper contains 4 sections, 13 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Definitions, Examples, and Induced Bilinear Forms
  3. Dual Representations
  4. $T^*$-extensions of quadratic Bol algebras

Key Result

Theorem 2.1

(KZ). Every two-dimensional left Bol algebra $(T, *, [\![, ,]\!])$ over $\mathbb{R}$ has a canonical basis $\{ e_1, e_2\}$ such that $(T, *, [\![, ,]\!])$ is of one of the following types: ${\mathrm{I.}}$$e_1 * e_2 = 0$, $[\![e_1,e_2 ,e_1]\!] = {\varepsilon}_2 e_2$, $[\![e_1,e_2 ,e_2]\!] = - {\varep

Theorems & Definitions (13)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Proposition 2.7
  • Lemma 3.1
  • Theorem 3.2
  • Proposition 3.3
  • ...and 3 more