Free Jordan Algebras and Representations of sl_2(J)

Abstract: Let J be a unital Jordan algebra and let sl_2(J) be the universal central extension of the Tits-Kantor-Koecher Lie algebra TKK(J). In part A, we study the category of (sl_2(J), SL_2(K))-modules. We characterize the dominant J-spaces, which are analogous to the classical notion of dominant highest weights. We also show some finitness results. For an augmented Jordan algebra J, we define in Part C the notion of smooth modules. We investigate the corresponding category in the spirit of Cline-Parshall-Scott highest weight categories. We show that the standard modules are finite dimensional if J is a finitely generated. Let J(D) be the free unital Jordan algebra over $D$ variables. The results of Part A suggest that the Lie algebra sl_2(J(D+1)) is FP_\infty. Similarly the results of part C suggest that the category of smooth sl_2(J(D))-modules is a generalized highest weight category. We show that both hypothesis implies the conjecture of [KM] about the dimensions of the homogenous components of $J(D)$. Surprisingly, the proofs of most results make use of some deep results of E. Zelmanov.