Structurable tori and extended affine Lie algebras of type BC\(_{1}\).
Extended affine Lie algebras were introduced, under a different name, by \textit{R. Høegh-Krohn} and \textit{B. Torresani} [J.~Funct. Anal. 89, 106--136 (1990; Zbl 0792.17019)] (See also \textit{B. N. Allison, S. Azam, S. Berman, Y. Gao} and \textit{A. Pianzola} [Mem. Am. Math. Soc. 603 (1997; Zbl 0879.17012)].) These algebras contain a large ideal, the core, whose quotient by its centre, called the centreless core, determines to a large extent the whole algebra. The centreless core is graded by a finite root system (not necessarily reduced), and thus can be described in terms of a coordinate nonassociative algebra. So far, these coordinate algebras had been determined for all possible root systems but BC\(_1\) and BC\(_2\). The BC\(_1\)-case is the object of study of this paper. It is proven than the coordinate algebras in this case are the so called structurable tori. These are structurable algebras \((A,{}^-)\) graded over a free abelian group \(\Lambda\) of finite rank: \(A=\oplus_{\sigma\in \Lambda}A_\sigma\), such that for any \(\sigma\in\Lambda\), \(\overline{A_\sigma}=A_\sigma\), \(\dim A_\sigma\leq 1\) and for any \(0\neq x\in A_\sigma\), there is an element \(y\in A_{-\sigma}\) such that \(xy=yx=1\) and \([L_x,L_y]=0=[R_x,R_y]\). Over fields of characteristic \(\neq 2,3\), structurable tori are shown to be divided naturally into three classes, according to the behavior of the subalgebra generated by the skew-Hermitian elements relative to the involution. Tori in one of the classes are completely classified, while in the other two classes this is done for \(\Lambda\) of rank \(2\). It turns out that all structurable \(2\)-tori are obtained from Hermitian forms over quantum \(2\)-tori with involution.
scientific article; zbMATH DE number 1993278
scientific article