A Class Of Representations Of The Skew Derivation Lie Algebra Over Quantum Torus

The derivation Lie algebras of quantum torus have many important application in the study of the representation theory of Lie algebras.The quantum torus contains the Laurent polynomial ring as its special case,and the derivation Lie algebra of the quantum torus contains the derivation Lie algebra of certain Jordan algebras as its subalgebras. Moreover,it is proved that the classifications of the integrable modules over the toroidal Lie algebras and some extended affine Lie algebras can be reduced to the classification of the modules over the derivation Lie algebras of the coordinate algebras.The structure and the representations of derivation Lie algebra of quantum torus have been studied extensively.In this paper,we focus on studying the representations of the skew derivation Lie algebra over the rank two quantum torus.We describe the results as follows:In chapter one,we recall the notion of skew derivation Lie algebra.In chapter two,we introduce a class of infinite-dimensional sl₂-modules and the functor F_g^α.Finally in chapter three,we study the structure of the image modules,under the functor F_g^α,of the infinite-dimensional sl₂-modules.Then we get a class of representations of the skew derivation Lie algebras with infinite-dimensional weight spaces.