Non-weight Representations Over Several Classes Of Lie Algebras

Lie algebra theory was founded in the nineteenth century later, and it has universally been used in many branches of mathematics and many subjects of physics. The present paper mainly studies the non-weight representations over three classes of Lie algebras(the Virasoro algebra, Witt algebras, and special linear algebras).We give two classes of irreducible non-weight representations over the Virasoro algebra (?), and classify a class of non-weight modules. The first class of modules are tensor products Ω(λa)(?) M, where the irreducible module Ω(λ, a) is defined in [33], and the irreducible module M is a locally finite9(?)+(k)-module for some fixed positive integer k. By the irre-ducibility of this class of modules we also determine the necessary and sufficient conditions for a class of (?)-modules Indeθ,λ(Bs(n), n∈Z+to be irreducible. The second non-weight modules are obtained by taking the tensor products of a finite number of irreducible modules Ω(λi, ai),1≤i≤m and the irreducible module M where M is also a locally finite (?)+(k)-module for some fixed positive integer k. By the irreducibility of this class of (?)-modules we also determine the necessary and sufficient conditions for another class of93-modules Indθ,λ1,λ2(Bs(n), n∈Z+to be irreducible. Moreover, we also classify the (?)-module struc-tures on C [do].From the irreducible modules over Weyl algebras Kn,n>1, using the "twisted" tech-nique, we obtain a class of Wn-modules. We also determine the necessary and sufficient conditions for this class of Wn-modules to be irreducible. This class of Wn-modules consist of weight and non-weight modules, and the non-weight modules are new. We also classify the Wn-module structures on the universal enveloping algebra of the Cartan subalgebra of Wn.Since the special linear algebra sln+1(C), n>1can be embedded in Witt algebra Wn, each Wn-module can be seen as an sln+1(C)-module. Then we obtain a class of new non-weight sln+i(C)-modules which are Wn-modules.The present paper consists of five chapters:Chapter1introduces the background of the paper, related definitions and known realities, and lists the main results of the paper.Chapter2studies the non-weight representations over the Virasoro algebra, gives two classes of non-weight modules, and classifies a class of non-weight modules.Chapter3obtains a class of irreducible representations over Witt algebras Wn, n>1, and classifies a class of non-weight modules.Chapter4obtains a class of new irreducible non-weight modules over the special linear algebras sln+1(C).In chapter5, some problems for further studies are listed.