Remarks on quadratic left Bol algebras

TL;DR

This paper explores quadratic left Bol algebras, providing examples, analyzing their structure, dual representations, and introducing T*-extensions, with a focus on their relation to quadratic Lie triple systems.

Contribution

It introduces the concept of T*-extensions for quadratic Bol algebras and investigates dual representations, especially coadjoint representations.

Findings

Two-dimensional quadratic real Bol algebras are quadratic Lie triple systems

Examples of quadratic Bol algebras are constructed and analyzed

Dual representations and T*-extensions are developed for quadratic 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.