realization of locally extended affine lie algebras of type $a_1$

Realization of locally extended affine Lie algebras of type $A_1$

Locally extended affine Lie algebras were introduced by Morita and Yoshii in [J. Algebra 301(1) (2006), 59-81] as a natural generalization of extended affine Lie algebras. After that, various generalizations of these Lie algebras have been investigated by others. It is known that a locally extended affine Lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight...

Multiloop Realization of Extended Affine Lie Algebras and Lie Tori

An important theorem in the theory of infinite dimensional Lie algebras states that any affine Kac-Moody algebra can be realized (that is to say constructed explicitly) using loop algebras. In this paper, we consider the corresponding problem for a class of Lie algebras called extended affine Lie algebras (EALAs) that generalize affine algebras. EALAs occur in families that are constructed from...

Extended affine Lie algebras and other generalizations of affine Lie algebras – a survey

Motivation. The theory of affine (Kac-Moody) Lie algebras has been a tremendous success story. Not only has one been able to generalize essentially all of the well-developed theory of finite-dimensional simple Lie algebras and their associated groups to the setting of affine Lie algebras, but these algebras have found many striking applications in other parts of mathematics. It is natural to as...

Anyonic Realization of the Quantum Affine Lie Algebras

We give a realization of quantum affine Lie algebras Uq(ÂN−1) and Uq(ĈN ) in terms of anyons defined on a one-dimensional chain (or on a two-dimensional lattice), the deformation parameter q being related to the statistical parameter ν of the anyons by q = eiπν . In the limit of the deformation parameter going to one we recover the Feingold-Frenkel [1] fermionic construction of undeformed affin...

Lie Algebras of Finite and Affine Type

Isotopy for Extended Affine Lie Algebras and Lie Tori

Centreless Lie tori have been used by E. Neher to construct all extended affine Lie algebras (EALAs). In this article, we study isotopy for centreless Lie tori, and show that Neher’s construction provides a 1-1 correspondence between centreless Lie tori up to isotopy and families of EALAs up to isomorphism. Also, centreless Lie tori can be coordinatized by unital algebras that are in general no...