Lie theory for symmetric Leibniz algebras

Mamuka Jibladze,Teimuraz Pirashvili

doi:10.1007/s40062-019-00248-x

Mamuka Jibladze, Teimuraz Pirashvili

Open Access

https://doi.org/10.1007/s40062-019-00248-x

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Abstract

Lie algebras and groups equipped with a multiplication mu satisfying some compatibility properties are studied. These structures are called symmetric Lie mu -algebras and symmetric mu -groups respectively. An equivalence of categories between symmetric Lie mu -algebras and symmetric Leibniz algebras is established when 2 is invertible in the base ring. The second main result of the paper is an equivalence of categories between simply connected symmetric Lie mu -groups and finite dimensional symmetric Leibniz algebras.

Highlights

One of the leading motivations for introducing Leibniz algebras by Loday was an intriguing possibility to define some mythical objects, which he named coquecigrues. Those should stand in the same relation to groups, as Leibniz algebras are to Lie algebras
The second author looked into his very old file of papers and discovered our notes written in 1995 and entitled “The first step to coquecigrue: the grin”, as well as unpublished notes by Ronco [9] written about the same time, where among other results our Corollary 6 is stated
We describe the theory of symmetric Leibniz algebras as a linear extension of the theory of Lie algebras

Summary

Introduction

One of the leading motivations for introducing Leibniz algebras by Loday was an intriguing possibility to define some mythical objects, which he named coquecigrues (after Rabelais). Those should stand in the same relation to groups, as Leibniz algebras are to Lie algebras. Dedicated to the memory of Jean-Louis Loday Reading these notes after 24 years, we think they still are of some interest. we introduce symmetric Leibniz algebras and state some of their properties that we will need. we define the closely related symmetric Lie μ-algebras, and in Sect. Let us finish the introduction with acknowledging very useful advice of the referee which helped to improve the paper a lot

Symmetric Leibniz algebras

Symmetric Lie -algebras

The case when 2 is invertible in K

Lie 6 -groups

Findings

Interpretation in language of algebraic theories