Lie tori -- a simple characterization of extended affine Lie algebras
type: Item
instance of: scholarly article
title: Lie tori -- a simple characterization of extended affine Lie algebras
zbMATH Open document ID: 1148.17017
author: Yoji Yoshii
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2007-02-01T00:00:00Z
review text: 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 (\(n\geq 0\)).
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17D99
zbMATH DE Number: 5120439
zbMATH Keywords: Lie torus
zbMATH Keywords: extended affine Lie algebra
zbMATH Keywords: core
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1166642158
OpenAlex ID: W2010112454
cites work: A class of nonassociative algebras with involution containing the class of Jordan algebras
cites work: Construction of extended affine lie algebras by the twisting process
cites work: Extended affine Lie algebras and their root systems
cites work: Lie algebras graded by the root systems 𝐵𝐶ᵣ,𝑟≥2
cites work: A characterization of affine Kac-Moody Lie algebras
cites work: Isotopes of some nonassociative algebras with involution
cites work: The root system and the core of an extended affine Lie algebra.
cites work: Structurable tori and extended affine Lie algebras of type BC\(_{1}\).
cites work: Derivations and Invariant Forms of Lie Algebras Graded by Finite Root Systems
cites work: Quantum tori and the structure of elliptic quasi-simple Lie algebras
cites work: The Alternative Torus and the Structure of Elliptic Quasi-Simple Lie Algebras of Type A 2
DOI: 10.2977/PRIMS/1166642158
description: scientific article; zbMATH DE number 5120439