Locally extended affine Lie algebras

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.