TKK algebras and vertex operator representations

Let be the Jordan algebra constructed from a semilattice S of ν (v ≥ 1) [see Allison et al., Mem. Amer. Math. Soc., 603 (1997)]. Let be the Lie algebra obtained from the Jordan algebra by the Tits-Kantor-Koecher construction. The TKK algebra is defined to be the universal central extension of the Lie algebra . In this paper we present a complete description of the TKK algebra , where S is the smallest possible (nonlattice) semilattice, which then allows us to give a representation to this Lie algebra by vertex operators.