Lie Tori—A Simple Characterization of Extended Affine Lie Algebras

Abstract

We show the existence of a nonzero graded form on a Lie torus by the existence of a nonzero graded form on a structurable torus. This gives a simple characterization of the core of an extended affine Lie algebra (EALA). Namely, the core of any EALA is a Lie torus, and any centreless Lie torus is the centreless core of some EALA. We also show that a graded form on a Lie torus is unique up to scalars.

The centroid of extended affine and root graded Lie algebras

“…It is known that K is a Lie torus with Z n (see [38,Corollary 7.3] for F = C or [25, Proposition 3(a)] for arbitrary F). Moreover, by [25,Theorem 6], K is obtained from the centreless Lie torus L = K/Z(K) by the construction of Proposition 4.11.…”

Locally extended affine Lie algebras

“…More generally, from Jordan or structurable Λ-tori ( [4,25] or [27]), where Λ is a torsion-free abelian group, one can construct various new EALAs.…”

Lie Tori and Their Fixed Point Subalgebras

“…Finally, using (LT2)(ii), one can check that L has the division property. Therefore, L is a Λ-Lie torus of type ∆ in the sense of [14].…”