Localization and filtration in extended affine Lie algebras

Abstract

We investigate the notions of \emph{localization} and \emph{filtration} in the context of extended affine Lie algebras. Our primary objective is to develop a localization theory that facilitates the construction of meaningful local substructures, particularly local affine Lie subalgebras. These subalgebras play a crucial role in understanding the global structure of extended affine Lie algebras. It is noteworthy that the existence of appropriate local subalgebras, particularly affine Lie subalgebras, is also fundamental to the representation theory of extended affine Lie algebras. As a natural outcome of our localization approach, we also introduce a formal notion of filtration for a given extended affine Lie algebra. This study is motivated by our interest in modular theory, specifically the integral structures of extended affine Lie algebras.