TKK algebras and vertex operator representations
Let \(\mathcal J(S)\) be the Jordan algebra constructed from a semilattice \(S\) of \(\mathbb R^\nu\) [see \textit{B. Allison, S. Azam, S. Berman, Y. Gao} and \textit{A. Pianzola}, Extended affine Lie algebras and their root systems, Mem. Am. Math. Soc. 603 (1997; Zbl 0879.17012)] and let \(\mathcal G(\mathcal J (S))\) be the Lie algebra obtained from the Jordan algebra \(\mathcal J(S)\) by the Tits-Kantor-Koecher construction [\textit{N. Jacobson}, Structure and representations of Jordan algebras, AMS Colloq. Publ. 39 (1968; Zbl 0218.17010)]. The TKK algebra \(\widehat{\mathcal G}(\mathcal J (S))\) is defined to be the universal central extension of the Lie algebra \(\mathcal G(\mathcal J (S))\). In the paper under review the author presents a complete description of the TKK algebra when \(S\) is the smallest possible (nonlattice) semilattice, the union of cosets \(2\mathbb Z^2\), \((1,0)+2\mathbb Z^2\) and \((0,1)+2\mathbb Z^2\) in \(\mathbb Z^2\). This TKK algebra is spanned by the elements \(x_\pm(\sigma)\), \(\alpha(\tau)\) and \(C_i(\rho)\), for \(i=1,2\), \(\sigma\in S\), \(\tau\in \mathbb Z^2\), and \(\rho\in 2\mathbb Z^2\). By using this basis the author constructs a representation of this Lie algebra by vertex operators which act on a vector space of mixed boson-fermion states.
scientific article; zbMATH DE number 1251305
scientific article