代数表示论的某些新进展

Some New Advances in the Representation Theory of Algebras

代数表示理论是代数学的一个新的重要分支．在近二十五年的时间里，这一理论有了很大的发展．关于代数表示的基础理论的介绍可参见文献［１０１］．本文主要从Ｈａｌ代数和拟遗传代数两个方面介绍代数表示论的一些最新进展．第一章给出了Ｈａｌ代数的基本理论及其方法，并且着重指出了利用这一理论和方法通过代数表示论去实现Ｋａｃ－Ｍｏｏｄｙ李代数及相应的量子包络代数．第二章介绍了拟遗传代数及其表示理论，以及这一理论与复半单李代数及代数群的表示理论等的联系．

The theory of representations of finite-dimesional algebras is one of the important branches of algebra and has developed rapidly during the last twenty-five years. The present paper is a survey of some new advances in this theory in view of Hall algebras and quasi-hereditary algebras. The paper is organized as follows. In section 1 we introduce the theory of Hall algebras and point out that in terms of the representation theory of algebras, the Hall algebra approach yields a realization of Kac-Moody Lie algebras and the quantization of the correspnoding universal enveloping algebras. Section 2 is devoted to survey the representation theory of quasi-hereditary algebras in connection with that of semisimple complex Lie algebras and algebraic groups