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

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.