POWRE: New Constructions for Quantized and Classical Enveloping Algebras

这项工作是通过妇女在研究和教育领域的职业机会方案作为一个研究/教育促进项目资助的。 这个项目有两个部分。 首先是量子化包络代数的Harish-Chandra模理论的发展。 这些模是重要的，因为它们对李群表示的研究的影响。 在经典情形下，半单李代数的Harish-Chandra模在作用限于极大紧李子代数时表现良好。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，基本结果，如量子哈里什-钱德拉的子商定理和有限性条件的同构组件将证明相对于这些子代数。 第二个主题涉及的推广经典定理由于Chevalley：不变量的对称代数的半单李代数下的伴随群survival到Weyl群不变量的对称化Cartan子代数。研究人员将研究这种设置的一种变化，其中两个群被替换为与给定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。 第一个不变子代数的象在第二个不变子代数中有有限余维数到什么程度？答案将把最近的解决方案的问题瓦拉赫在一般情况下。 主要的技术是利用所谓的Letzter映射，最初开发的量子化包络代数。

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.