On representations of Bol algebras

TL;DR

This paper introduces the concept of representations for Bol algebras, extending classical theorems like Engel's and Ado-Iwasawa, and classifies regular representations for two-dimensional cases.

Contribution

It defines Bol algebra representations, proves analogues of key theorems, and classifies regular representations for all two-dimensional Bol algebras.

Findings

Representation category is a tensor category

Analogues of Engel's and Ado-Iwasawa theorems established

Complete classification of regular two-dimensional Bol algebra representations

Abstract

In this paper, we introduce the notion of representation of Bol algebra. We prove an analogue of the classical Engel's theorem and the ex- tension of Ado-Iwasawa theorem for Bol Algebras. We study the category of representations of Bol algebras and show that it is a tensor category. In the case of regular representations of Bol algebras, we give a complete classification of them for all two-dimensional Bol algebras.