Categories of Jordan Structures and Graded Lie Algebras

TL;DR

This paper explores the relationship between Z-graded Lie algebras and Jordan structures, providing characterizations and functorial constructions that link these algebraic categories.

Contribution

It introduces a functorial modification of the TKK construction to relate Z-graded Lie algebras with Jordan pairs, triples, and algebras.

Findings

Z-graded Lie algebra corresponds to Jordan pair if generated by odd components and second homology is trivial

Provides categorical equivalences between Jordan structures and certain Z-graded Lie algebras

Extends similar descriptions to Jordan triple systems and Jordan algebras

Abstract

In the paper we describe the subcategory of the category of Z-graded Lie algebras which is equivalent to the category of Jordan pairs via a functorial modification of the TKK construction. For instance, we prove that a Z-graded Lie algebra can be constructed from a Jordan pair if and only if it is generated by odd graded components and the second graded homology group is trivial. Similar descriptions are obtained for Jordan triple systems and Jordan algebras.