Let J be a unital Jordan algebra, and let \(\widehat{\mathfrak {sl}}_2(J)\) be the universal central extension of the Tits-Kantor-Koecher Lie algebra. In Part A, we study the category of \((\widehat{\mathfrak {sl}}_2(J), \textrm{SL}_2(\mathbb {K}))\) -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 \(\hbox {Ext}\) -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 (Kashuba and Mathieu, Adv. Math. 383(35), 107690 2021) and (Dotsenko and Hentzel, 2025) 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 \(D\ge 2\) . Surprisingly, the proofs of most of these results make use of deep theorems of E. Zelmanov.
