Multi-affine Lie algebras and their representations

广义仿射李代数是仿射Kac-Moody李代数的自然推广。它们是数学物理学家在理解高维弦理论和共形场论时引入的。 这类李代数与Saito的扩展仿射根系、Slodowy的广义交矩阵李代数以及Berman-Moody、Benkart-Zelmanov、Neher等人研究的根分次李代数密切相关。每个扩展仿射李代数都与一个有限的不可约根系（可能是非约的）和一个称为零度的非负整数相关联。零值 得到有限维单李代数，而零维李代数得到仿射Kac-Moody李代数。大于1的零度产生了多仿射李代数，其中包括许多有趣的例子，如环面李代数，以及某些与量子环面甚至非结合环面协调的李代数。扩展仿射李代数的结构理论已经很好地理解了。然而，扩张仿射李代数的表示理论是困难的，远未完成。在这个计划中，我们将继续研究扩展仿射李代数，它们的表示和量子化。 我们打算探索一些相关的代数，如多仿射李超代数以及某些GIM李代数。以下三个主题是我们短期或长期的兴趣。**

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.***************************