Representations of Infinite-dimensional Lie Algebras

摘要：Mukhine.Mukhin研究了仿射Lie代数的表示理论及其在多个方向上的量子化，包括仿射Kac-Moody代数(与B.Feigin，M.Jimbo和T.Miwa)的可积表示的协不变量理论，量子仿射群的有限维表示理论。Frenkel)和由Knizhnik-Zamolodchikov定义的Lie代数和量子群的表示范畴之间的函子以及量子化的Knizhnik-Zamolodchikov方程(与A.Varchenko)之间的函子。该项目有望描述与复半单李代数及其仿射和量子类似的表示理论相关的美丽的代数-组合结构的许多特征。特别是，这项研究涉及到许多数学领域，如对称函数理论和Young表组合理论、特殊函数理论、精确可解格子模型、保形场理论、1+1维质量场理论等，因为所有这些理论都描述了不同类型的李代数的对称性。

AbstractMukhinE. Mukhin studies the representation theory of affine Liealgebras and their quantizations in several directions including thetheory of coinvariants of integrable representations of affineKac-Moody algebras (with B. Feigin, M. Jimbo and T. Miwa), the theoryof finite-dimensional representations of quantum affine groups (withE. Frenkel) and the functors between categories of representations ofLie algebras and quantum groups defined by Knizhnik-Zamolodchikov andquantized Knizhnik-Zamolodchikov equations (with A. Varchenko). The project is expected to result in description of many features ofthe beautiful algebraic-combinatorial structure related torepresentation theory of complex semi-simple Lie algebras and their affine andquantum analogs. In particular, this research has connections to many areas ofmathematics such as theory of symmetric functions and combinatorics ofYoung tableaux, theory of special functions, exactly solvable latticemodels, conformal field theory, massive field theory in 1+1dimensions, etc; as all these theories have symmetries described interms of Lie algebras of different kind.