Crystal Bases And Canonical Bases Of Quantum Groups

This thesis investigates the theory of crystal base of quantum group as well as its applications to canonical base theory.For any symmetrizable Kac-Moody algebra g, a path realization of B(A) was in-troduced by Littelmann, where B(A) is the crystal base of irreducible integrable highest weight representation (with a dominant highest weight A) of quantum group Uq(g). To generalize this result, an explicit realization of B(∞), the crystal base of the negative part of Uq(g), is given in terms of equivalent classes of so called generalized Littel-mann's paths.Filtration or composition series of various tensor products of two Uq(g)-modules are also investigated in this thesis. Given a pair of dominant weights (λ,μ), Lusztig conjectured that for g of finite type, there exists a composition series of Up(g)-module V(λ) (?) V(-μ) compatible with the canonical base. We extend this to a more general form and prove this extended conjecture. Namely, for any symmetrizable Kac-Moody algebra g, there is a composition series of the Uq(g)-module V(λ) (?) V(μ) compatible with the canonical base. In the same manner, some filtration of V(λ) (?) V(-μ) is also constructed compatible with the canonical base and the successive quotients are either zero or irreducible integrable highest weight Uq(g)-modules. In particular, one can ob-tain a composition series by deleting the superfluous terms in the filtration. Similarly, a nice composition series can be constructed whenλ-μis of a negative level. Further-more, when g is of affine type and y is of level 0, we construct a composition series of V(λ) (?) Vmax(γ) compatible with the canonical base, which provides a representation-theoretic interpretation to the decomposition of crystal base.Finally for g of any type, one can use the filtration of V(λ)(?) V(-μ) to define some ideal U' of the modified quantized enveloping algebra U, where U' consists of all elements in U which annihilate all irreducible integrable highest weight Uq(g)-modules. It is shown that the quotient algebra (?) inherits canonical base as well as an analogue of comultiplication from U. One can also prove that (?) is, in some sense, dual to the quantum coordinate ring defined by Kashiwara, which generalizes the result of Lusztig's for g of finite type.