| This dissertation is devoted to the study of the algebra Uqg + , which occurs as the positive part in the standard triangular decomposition Uqg -⊗U0qg ⊗Uqg + of the quantized enveloping algebra Uqg associated with a finite-dimensional complex semisimple Lie algebra g . We view Uqg + as a one-parameter deformation of the universal enveloping algebra Ug+ of a maximal nilpotent subalgebra g+ of g .;It is assumed throughout that the base field K has characteristic 0, and that q∈K is transcendental over the subfield Q of rational numbers.;In Chapter 2, we classify all finite-dimensional simple Uqg + -modules for g an arbitrary finite-dimensional complex semisimple Lie algebra. The classification is accomplished without any assumption on the base field, but if K is algebraically closed, all such simple modules are shown to be one-dimensional. Some necessary conditions for a finite-dimensional Uqg + -module to be indecomposable are obtained as well.;The center of Uqg + is discussed in Chapter 3, and in Chapter 4, we prove a quantum version of Kostant's well-known separation of variables for the algebra Uqsln+1 + , where sln+1 is the simple Lie algebra of all (n+1) x ( n+1) complex matrices of trace 0. To be precise, we show that Uqsln+1 + is free as a module over its center. This result is subsequently applied to the study of infinite-dimensional Uqsln+1 + -modules.;Finally, in Chapter 5 we obtain results about the primitive ideals of Uqsln+1 + that sharply contrast with known results of Dixmier for the universal enveloping algebra of a finite-dimensional nilpotent Lie algebra. We construct primitive quotients of Uqsln+1 + that are simple Noetherian domains of even Gelfand-Kirillov dimension, but which are not isomorphic to a Weyl algebra. Also, we show that there exist primitive ideals of Uqsln+1 + which are not maximal ideals.;We conclude this thesis with a conjecture: the number of orbits of primitive ideals of Uqg + under the action of an appropriately chosen algebraic torus is equal to the order of the Weyl group of g . This conjecture is verified for some particular choices of g . |
