Free Jordan Algebras and Representations of $\widehat{\mathfrak{sl}}_2(J)$

Let $J$ be a unital Jordan algebra, and let $\widehat{\mathfrak{sl}}_2(J)$ be the universal central extension of its Tits-Kantor-Koecher Lie algebra. In Part A, we study the category of $(\widehat{\mathfrak{sl}}_2(J), SL_2(K))$-modules. We characterize the dominant $J$-spaces, which are analogous to the dominant highest weights appearing in classical settings. A family of universal envelopes $\mathcal{U}_n(J)$ associated to such modules is introduced and studied. We also prove some finiteness theorems. In Part C, we define the notion of smooth $\widehat{\mathfrak{sl}}_2(J)$-modules for augmented Jordan algebras $J$, and investigate the category of smooth modules in the spirit of Cline-Parshall-Scott highest weight categories. We show that the standard modules of this category are finite dimensional when $J$ is finitely generated. The free unital Jordan algebra $J(D)$ over $D$ variables is an elusive object, but finiteness and Ext-vanishing properties suggest that the smooth $\widehat{\mathfrak{sl}}_2(J(D))$-modules with even eigenvalues might form a generalized highest weight category. However, we prove that such an assertion would contradict recently obtained information about the growth of free Jordan algebras. See [24] and [13] for more details. It then follows that the category of smooth $\widehat{\mathfrak{sl}}_2(J(D))$-modules with even eigenvalues is not a generalized highest weight category when $D\geq 2$. Surprisingly, the proofs of most of these results make use of deep theorems of E. Zelmanov.