Lie tori -- a simple characterization of extended affine Lie algebras
Extended affine Lie algebras (or EALAs for short) constitute a class of complex Lie algebras that includes the finite dimensional simple Lie algebras and the affine Lie algebras (see the monograph by \textit{B. Allison, S. Azam, S. Berman, Y. Gao} and \textit{A. Pianzola} [Mem. Am. Math. Soc. 603 (1997; Zbl 0879.17012)]). To a great extent, any such Lie algebra is determined by its so called centerless core, which is a loop algebra in the affine case. The paper under review is devoted to characterize these centerless cores in an intrinsic way. To this aim, given an abelian group \(G\) and a (not necessarily reduced) root system \(\Delta\), the concept of a Lie \(G\)-torus of type \(\Delta\) is defined as a Lie algebra satisfying several axioms fulfilled by the cores of EALAs. These axioms express that the Lie torus \(L\) is both graded over \(G\) and over the root system \(\Delta\) in a compatible way, so that \(L=\bigoplus_{\mu\in \Delta\cup\{ 0\}}\bigoplus_{g\in G} L_\mu^g\), plus a ``division property'' and the condition \(\dim L_\mu^g\leq 1\) for any \(g\in G\) and \(\mu\in\Delta\). Then, starting with a proof of the existence of suitable symmetric invariant bilinear forms on structurable tori, which are nonassociative algebras that coordinatize the Lie tori of type \(BC_1\), the author proves the existence of nonzero ``graded'' symmetric invariant bilinear forms on Lie \(G\)-tori if \(G\) is torsion free. This result allows him to characterize the centerless cores of EALAs precisely as the centerless Lie \({\mathbb Z}^n\)-tori (\(n\geq 0\)).
scientific article; zbMATH DE number 5120439