The cores of extended affine Lie algebras of reduced types were classified except for type A 1 . In this paper we determine the coordinate algebra of extended affine Lie algebras of type A 1 . It turns out that such an algebra is a unital n -graded Jordan algebra of a certain type, called a Jordan torus. We classify Jordan tori and get five types of Jordan tori.
We propose a new simplified definition of extended affine Lie algebras (EALAs for short), and also discuss a general version of extended affine Lie algebras, called locally extended affine Lie algebras (local EALAs for short). We prove a conjecture by V. Kac for local EALAs. It turns out that the root system of a local EALA becomes a locally finite version of an extended affine root system. Several examples of new EALAs and local EALAs are introduced, and finally we classify local EALAs of nullity 0 and show the connection to locally finite split simple Lie algebras.
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.
Abstract. We consider a natural generalization of both locally finite irreducible root systems and extended affine root systems defined by Saito. We classify the systems.
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