量子群及表示的范畴理论和Yang-Baxter方程的解

Abstract：In this project, we mainly studied the structure and epresentation of quantum groups and singular solutions of Yang-Baxter Equation through the algebraic method, which.background is in the theory of algebraic deformations and the meaning of quantum groups in physics. The main contents include as follows: the constructure of weak Hopf algebra with one parameter and singular solutions of Yang-Baxter Equation; the structure of quantum quasi-doubles and the categorical theory of their representations; the characterizations of quasi-(co-)braided bialgebras and FRT-Constructures; An improvement of crossed products of Hopf algebras; the.structure of Hopf algebras; fuzzy sub-groupoids. Our major results and their meanings are as follows: getting the constructure of the quasi-quantum enveloping algebras wslq(2) and vslq(2), and moreover from them to obtain singular regular solutions with a parameter of the quantum Yang-Baxter equation；giving a sufficient and.necessary condition for a double biproduct to become a Hopf algebra in a braided tensor category; finding the relation between the quantum double of a Clifford monoid and the quantum doubles of the groups which constructing the Clifford monoid; regular solutions of Yang-Baxter equation are constructed from every quasi-(co-)braided almost bialgebra, conversely, by FRT- constructures,.it is shown that every (singular) solution of Yang-Baxter equationcan be built from a quasi-cobraided bialgebra; the Jacobson radical of a twisted graded algebra is proved to be a graded ideal, then two Fisher’s questions are solved; the homological dimensions of crossed products are studied and it is shown that when H is a finite dimensional semisimple and cosemisimple.Hopf algebra, the weak and global dimensions of R and R#σH are equal to each other; the structure theorem of a right Hopf module Mover a right Hopf algebra H is given as: M.(N.K)⊕(M/Imα), and using of it, a sufficient and necessary is got under which H*rat becomes a right H-Hopf module; a general theory on fuzzy subgroupoids is developed with respect to t-norm, and in particular, a characterization of a fuzzy sub-groupoid, which is induced by a probability space, is given.

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