量子环面上一些导子李代数的表示

For the derivation Lie algebra over a torus with n variables(n>1), Billig and Futorny have lately classified all the irreducible modules with finite dimensional weight spaces, one class of which is the module of tensor fields, the other is the so-called module of highest weight type. The construction of the module of highest weight type is closely related to a subalgebra, which is isomorphic to the direct sum of a torus with n-1 variables and its derivation Lie algebra. Quantum torus is a non-commuting generalization of torus. For the derivation Lie algebras over quantum tori, the construction is known for the modules of tensor fields, but not for the module of highest weight type, since the structure and representations of the subalgebra related to the construction in this case is not clear. One purpose of this project is to construct the modules of highest weight type for the derivation Lie algebras over rational quantum tori, and furthermore to study the properties of general irreducible modules with finite dimensional weight spaces. The other purpose is to study the module of tensor fields for the skew-derivation Lie algebras over rational quantum tori. This project should make some contribution to the classification of irreducible modules with finite dimensional weight spaces for the derivation Lie algebras, and representations of the skew-derivation Lie algebras, over rational quantum tori.

对多变量交换环面上全导子李代数，最近Billig和Futorny给出了其权空间有限维不可约模的完全分类结果，一类是张量场模；另一类是高权型模。其高权型模的构造涉及到了它一个子代数的表示，这个子代数同构于少一个变量的交换环面与其全导子李代数的直和。量子环面是多变量交换环面的一个(非交换)推广，其全导子李代数的张量场模已有构造，但是高权型模并没有构造，主要是由于涉及到的子代数的结构和表示并不明了。在本项目中，我们的目的一个是对有理量子环面构造其全导子李代数的高权型模，进而可以研究其一般权空间有限维不可约模的性质；另一方面我们还会研究有理量子环面的斜导子李代数的张量场模的性质。本项目对有理量子环面上的导子李代数的权空间有限维不可约模的分类工作，及其斜导子李代数的表示的研究，必然会有一定的贡献。