Localization and filtration in extended affine Lie algebras
Saeid Azam
TL;DR
This work develops a localization framework for extended affine Lie algebras that yields local affine substructures while preserving core data, enabling localized analyses of structure and representation theory. It builds on Neher’s construction and affinization to produce canonical Lie covers of minimal dimension and establishes precise tameness criteria for localized subalgebras. A core-based localization complements these results by isolating isotropic interactions and ensuring non-degeneracy under suitable conditions. The paper further introduces an ascending filtration of the root system and the corresponding $EALA$ chain, providing a practical reduction mechanism to study higher-nullity algebras via lower-nullity analogs, and illustrates the theory with explicit toroidal and $D_4$-inspired examples.
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.