无穷维李理论及其应用

本项目研究仿射Kac-Moody李代数和它的高维推广及相关问题。单位圆圈（环面）到紧李群的所有光滑映射的集合就是圈群（双圈群），对应的李代数就是圈代数（双圈代数），即无中心仿射Kac-Moody李代数（零度为2的无中心高维仿射李代数）。具体问题如下：1.零度为2的量子化高维仿射代数的表示理论，这类代数是由Ginzburg-Kapranov-Vasserot在研究代数曲面的Langlands互反性时得到的；2.与Novikov代数相关的量子顶点代数，量子顶点算子代数是由Etingof-Kazhdan发展起来的；3.高维仿射李代数坐标代数的低阶（斜）二面体同调群，带对合的结合代数的（斜）二面体同调是由Loday等人最先提出的，它是Connes循环同调直和分解的两个直和项；4.圈群的相关几何问题，圈群上类似于左不变Einstein度量的存在性及圈群Toda系统解的性质。

This project investigates affine Kac-Moody Lie algebra, its high dimensional generalization and related topics. All smooth maps from unit circle to a compact Lie group form a loop group, and the corresponding Lie algebra is the loop algebra which is the centerless affine Kac-Moody Lie algebra. All smooth maps from the torus (product of two unit circles) to a compact Lie group form a double loop group, the corresponding Lie algebra is the double loop algebra which is the centerless extended affine Lie algebra of nullity two. More precisely: (1) the representation theory of quantized extended affine algebra of nullity two, this algebra was introduced by Ginzburg-Kapranov-Vasserot in the study of the Langlands reciprocity for algebraic surfaces; (2) quantum vertex algebra associated with Novikov algebra, quantum vertex operator algebra was developed by Etingof-Kazhdan; (3) (skew) dihedral homology of coordinate algebras of extended affine Lie algebras, (skew) dihedral homology of involutive associative algebras was first introduced by Loday et al, it is the two direct summands of Connes cyclic homology; (4) geometric problems related to loop groups, one is the existence of analogue of left-invariant Einstein metrics on loop groups and the other is the structure of solutions for Toda systems associated to loop groups.