Representations Over Kac-Moody Algebras And Their Quantum Groups

Let g be a symmetrizable Kac-Moody algebra over C with a fixed Cartan subalgebra h,and let U=Uq(g)be the quantized enveloping algebra associated to g with its Cartan part U0.Nilsson[61,62]studied the category of all U(g)-modules whose restriction to U(h)are free of finite rank when g is of finite type.He proved that this category is not empty if and only if g is of type An(n≥1)or Cn(n≥2).In addition,all U(h)-free modules of rank 1 are constructed and their module structures are studied in[61,62].Motivated by Nilsson’s work,we first study a class of non-weight modules of the quantized enveloping algebra U associated to g of finite type in this thesis.More precisely,assume that q is not a root of unity,we investigate the category C of all U-modules which are finitely generated when restricted as modules over U0.Accordingly,we classify all Umodule structures on U0 with the action of U0 itself by its multiplication,i.e.,U0-free modules of rank 1.Our strategy to show the existence of U0-free modules of rank 1 is completely different from that in[62],and surprisingly,we obtain that such U-modules exist if and only if U is of type An(n≥1),Bn(n≥2)or Cn(n≥3).Moreover,all U0-free modules of rank 1 are constructed and their module structures are studied in each type.Secondly,our research concerns itself with the hom-space between parabolic Verma modules over any symmetrizable Kac-Moody algebra g.Assume that I={1,2,…,n}is an indexed set,where n is the rank of g.We extend the parabolic category Os over g first studied by Rocha-Caridi and Wallach[64]to general case in where the parabolic subalgebra can be chosen for any subset S of I.We introduce a reduction rule of homspaces between parabolic Verma modules over different Kac-Moody algebras which yields some applications on parabolic Verma module homomorphisms.