Coordinate algebras of extended affine Lie algebras of type \(A_1\)
The cores of extended affine Lie algebras of reduced types were classified except for type \(A_1\) [cf. \textit{S. Berman, Y. Gao} and \textit{Y. Krylyuk}, J. Funct. Anal. 135, 339-389 (1996; Zbl 0847.17009)]. In this paper the author first shows that the core of any extended affine Lie algebra of type \(A_1\) is a central extension of a Lie algebra \(\text{TKK} (J)\) (obtained from a Jordan algebra \(J\) by the Tits-Kantor-Koecher construction), where the so-called coordinate algebra \(J\) is a unital \(\mathbb Z^n\)-graded Jordan algebra of certain type, called a Jordan torus. The major part of the paper is devoted to a classification of Jordan tori (over any field of characteristic \(\neq 2\)) into five classes. In particular, a Jordan torus is either of Hermitian, Clifford, or Albert type, and for each type the author determines the possible Jordan tori.
extended affine Lie algebra of type \(A_1\)
scientific article; zbMATH DE number 1565856
scientific article