POWRE: New Constructions for Quantized and Classical Enveloping Algebras

这项工作由妇女在研究和教育中的职业机会(POWRE)方案作为研究/教育增强项目提供资金。这个项目包括两个部分。第一个是发展了量子化包络代数的Harish-Chandra模理论。这些模块很重要，因为它们对李群表示的研究有影响。在经典情形下，当作用限于极大紧李子代数时，半单李代数的Harish-Chandra模表现得很好。为了构建一个类似的量子理论，研究人员将分析这些特殊李子代数的已知量子类似物，并确定新的量子类似物。接下来，将证明与这些子代数有关的基本结果，如量子Harish-Chandra次商定理和关于同型分支的有限条件。第二个主题是Chvalley经典定理的推广：半单李代数的对称代数在伴随群满射下的不变量推广到对称化的Cartan子代数的Weyl群不变量上。研究人员将研究这种设置的一个变体，其中这两个群被与给定的半单李代数的某些李子代数相关联的较小的伴随群和Weyl群所取代。第一个不变子代数的像在第二个不变子代数中的余维有限到什么程度？答案将把沃拉赫的一个问题的最新解决方案放在一个普遍的背景下。主要技术是利用所谓的Letzter映射，它最初是为量化包络代数开发的。

This work is funded through the Professional Opportunities for Women in Research and Education (POWRE) program as a Research/Educational Enhancement Project. This project has two parts. The first is the development of a theory of Harish-Chandra modules for quantized enveloping algebras. These modules are important because of their impact on the study of Lie group representations. In the classical case, the Harish-Chandra modules of semisimple Lie algebras behave nicely when the action is restricted to maximal compact Lie subalgebras. To construct an analogous quantum theory, the investigator will analyze already known quantum analogs of these special Lie subalgebras and identify new ones. Next, basic results such as a quantum Harish-Chandra's subquotient theorem and a finiteness condition on isotypic components will be proved relative to these subalgebras. The second topic concerns generalizations of the classical theorem due to Chevalley: the invariants of the symmetric algebra of a semisimple Lie algebra under the adjoint group surjects onto the Weyl group invariants of the symmetrized Cartan subalgebra. The researcher will study a variation of this setup where the two groups are replaced with the smaller adjoint group and Weyl group associated to certain Lie subalgebras of a given semisimple Lie algebra. To what extent does the image of the first invariant subalgebra have finite codimension in the second? Answers will put recent solutions to a question of Wallach in a general context. The main technique is to exploit the so-called Letzter map, originally developed for quantized enveloping algebras.