Jordan algebras and weight modules

Abstract

We consider bounded weight modules for the universal central extension ${\mathfrak{sl}}_2(J)$ of the Tits-Kantor-Koecher algebra of a unital Jordan algebra $J$. Universal objects called Weyl modules are introduced and studied, and a combinatorial dominance criterion is given for analogues of highest weights. Specializing $J$ to the free Jordan algebra $J(r)$ of rank $r$, the category $\mathcal{C}^{fin}$ of finite-dimensional $\mathbb{Z}$-graded ${\mathfrak{sl}}_2(J)$-modules shares many properties with the representation theory of algebraic groups. Using a deep result of Zelmanov, we show that this subcategory admits Weyl modules. By analogy, we conjecture that $\mathcal{C}^{fin}$ is a highest weight category. The resulting homological properties would then imply cohomological vanishing results previously conjectured as a way of determining graded dimensions of free Jordan algebras.