广义Kac-Moody代数的包络代数的半典范基

The notion of quantum groups (or quantized universal enveloping algebras) was introduced independently by V. G. Drinfeld and M. Jimbo around 1985 in order to explain trigonometric R-matrices in 2-dimenisonal solvable models in statistical mechanics. As certain families of Hopf algebras, quantum groups are deformations of universal enveloping algebras of Lie algebras. They were first used to construct solutions to the quantum Yang-Baxter equation. Since then they have found numerous applications in areas ranging from theoretical physics to modular representations of reductive algebraic groups. The representation theory of quantum groups is one of the interesting topics over the past 20 years. In particular, the theory of crystal bases or canonical bases developed independently by M. Kashiwara and G. Lusztig provides a powerful combinatorial and geometric tool to study the representations of quantum groups. Around 1990, the canonical bases or global crystal bases of quantum groups were constructed independently by G. Lusztig and M. Kashiwara , which simultaneously give the canonical bases or global crystal bases of all irreducible highest weight integrable representations. Degenerate the parameter q to 1, we obtain the canonical bases or global crystal bases of the corresponding enveloping algebras. In 2000, via the notion of constructible functions Lusztig extended the definition of semicanonical bases in the general cases. . The generalized Kac-Moody algebras $\fkg$ were introduced by Borcherds in his study of Monstrous moonshine. The quantum version $\U_q(\fkg)$, now called quantum generalized Kac-Moody algebras, were constructed by Kang. Later, Jeong, Kang and Kashiwara introduced and developed crystal bases and global crystal bases for quantum generalized Kac--Moody algebras. Then Kang and Schiffmann gave a geometric construction of canonical bases and proved that when $a_{ii}\neq0$ for all $i\in I$, the canonical bases constructed by them coincide with the global crystal bases constructed by Jeong, Kang and Kashiwara. Similar results also were obtained by Li and Lin . Moreover, Joseph and Lamprou gave another realization of crystal bases of quantum generalized Kac-Moody algebras by Littelmann''s path model. Furthermore, Kang, Kashiwara and Schiffmann also gave a geometric construction of crystal bases. . The applicant and his cooperator have systematically studied the representations of modified quantum generalized Kac-Moody algebras and the fundamental representations of special type Borcherds-Cartan matrices. Based on the work before, this item is focused on the construction of the semicanonical bases of enveloping algebras of generalized Kac-Moody algebras.

广义Kac-Moody代数是Borcherds 在研究月光猜想(Monstrous moonshine) 的时候引入的一类新的无限维李代数，其量子形式（或者叫做量子广义 Kac-Moody代数）是Kang引入的. 后来，Jeong, Kang 和Kashiwara 研究了量子广义Kac-Moody 代数的表示理论，晶体基理论以及相应的整体晶体基. 另外Kang 和Schiffmann 用几何的方法实现了典范基. 同时，林和李也得到了相同的结论. 申请人与合作者已经研究了改进的(modified)量子广义Kac-Moody代数的表示及特殊的Borcherds-Cartan矩阵的基本表示。本项目主要研究广义Kac-Moody代数包络代数的半典范基的构造，进一步我们希望运用Kang, Kashiwara 和Schiffmann几何实现晶体基时引入的不可约分支去实现包络代数的半典范基.

本项目的研究目标分为两个部分，一是研究虚单根处的结构和表示，另一个是在此基础之上给出广义量子代数的一个自然的整形式，从而研究相应包络代数的半典范基。到目前为止，第一个目标已经完成，并且相应的成果已经投递到国内的相关杂志；第二个目标中已经给出了相应的整形式，关于其包络代数的几何结构还在继续研究中，本人在今后将继续此课题，希望在今年完成全部目标工作。