On the Tits-Kantor-Koecher construction of unital Jordan bimodules

TL;DR

This paper investigates the relationship between Jordan algebra representations and Lie algebra structures derived via the Tits-Kantor-Koecher construction, establishing functors and classifying specific algebra types with tame bimodule categories.

Contribution

It introduces adjoint functors connecting Jordan bimodules and graded Lie modules, and classifies Jordan algebras with Clifford type semisimple parts where bimodule categories are tame.

Findings

Constructed functors between Jordan bimodules and Lie algebra modules.

Classified Jordan algebras with Clifford type semisimple parts and tame bimodule categories.

Abstract

In this paper we explore relationship between representations of a Jordan algebra $\J$ and the Lie algebra $\g$ obtained from $\J$ by the Tits-Kantor-Koecher construction. More precisely, we construct two adjoint functors $Lie :\JJ\to \ggm$ and $Jor:\ggm\to\JJ$, where $\JJ$ is the category of unital $\J$-bimodules and $\ggm$ is the category of $\g$-modules admitting a short grading. Using these functors we classify $\J$ such that its semisimple part is of Clifford type and the category $\JJ$ is tame.