Multi-affine Lie algebras and their representations

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Extended affine Lie algebras are a natural generalization of the affine Kac-Moody Lie algebras. They were introduced by mathematical physicists in understanding the high dimensional string theory and conformal field theory. This class of Lie algebras is closely related to the extended affine root systems of Saito, the generalized intersection matrix  Lie algebras of Slodowy, and the root graded Lie algebras studied by Berman-Moody, Benkart-Zelmanov, Neher, among others. Every extended affine Lie algebra is associated with a finite irreducible root system (possibly non-reduced) and a nonnegative integer called nullity. The nullity zero yields exactly finite dimensional simple Lie algebras while the nullity one yields exactly affine Kac-Moody Lie algebras. The nullity greater than one yields multi-affine Lie algebras which include many interesting examples such as toroidal Lie algebras, and certain Lie algebras coordinated with quantum tori and even non-associative tori.*The structure theory of the extended affine Lie algebras has been well understood.*However, the representation theory of the extended affine Lie algebras is hard and far from complete. In this proposal, we will continue to study the extended affine Lie algebras, their representations and quantizations. We intend to explore some related algebras such as multi-affine Lie super-algebras as well as certain GIM Lie algebras. The following three topics are our interests for a short or longer term.***************************