Non-weight Modules Over Some Generalized Weyl Algebras

This thesis mainly focuses on three kinds of interesting generalized Weyl algebras,namely Rueda’s algebras,the algebras Uq(f(K))and the algebras Uq(f(K,H)),and the research scope mainly involves the constructions and classifications of some irreducible modules over these three kinds of algebras.Firstly,since these three kinds of algebras are generalized Weyl algebras,we review the definitions,properties and many specific examples of generalized Weyl algebras.Secondly,we consider two classes of non-weight modules over the Rueda’s algebras R=R(f,(?)):one is a class of R0-free module with rank 1 restricted to its "Cartan part" R0=C[H];the other is so-called Whittaker module in the sense of Kostant.For the case of free module with rank 1,we refer to Nilsson’s research method for sl2,and make full use of the special properties of the polynomial algebra C[H],all such modules over R are constructed,and a necessary and sufficient condition for determining the irreducibility of such modules is given.When such modules are reducible,we find all submodules.In particular,each reducible module has only finitely many maximal submodules and a unique simple submodule.As an application,we recover some results about U(h)-free modules over the Lie algebra sl2 obtained by J.Nilsson and over the Lie superalgebra osp(1|2)obtained by Y.Cai and K.Zhao.As for Whittaker modules over the Rueda’s algebras R,after referring to some methods of Kostant in dealing with complex simple Lie algebras,for the case whether(?)is a root of unit or not,we get the classification of all irreducible Whittaker modules for R.Finally,we hope to study the Whittaker modules over these three kinds of algebras in the framework of generalized Weyl algebras.Note that as early as 2009,Benkart and Ondrus have studied Whittaker modules over the generalized Weyl algebras,and one of the important conclusions is that there is a one-to-one correspondence between the isomorphic classes of Whittaker modules for generalized Weyl algebras A=R(φ,t)and φ-stable left ideals of base ring R.Therefore,we make use of Benkart-Ondrus’s conclusions to characterize φ-stable left ideals of these three kinds of algebras.Soon,we first acquire the previous results about Whittaker modules over the Rueda algebras.For the algebras Uq(f(K))and the algebras Uq(f(K,H)),we mainly describe the maximalφ-stable ideals of their base rings,hence simple Whittaker modules are also classified.In particular,we give an explicit description of the centers of these two kinds of algebras.