Semilattice Graded Weak Hopf Algebra Structure

Because of the important role of Hopf algebra in the theory of quantum group and related mathematical physics, along with the deepening of researches, the meaning of some weaker concepts of Hopf algebra is understood and is paid close attention more and more. A well-known example is weak Hopf algebra, which is introduced in [L2] for studying the non-invertible solution of Yang-Baxter Equation based on this class of bialgebras (in [L2] and [L6]). Due to the importance of Yang-Baxter Equation in theoretical physics, its solution is the keystone in research. The theory of singular solutions extends largely the scope of the research field. On the other aspect, there is a tight relation between weak Hopf algebra and regular monoid, for example, a semigroup algebra is a weak Hopf algebra if and only if the semigroup is a regular moniod. Obviously, it is necessary to find more non-trivial weak Hopf algebras in order to study these two aspects deeply.In Chapter 1, we construct a so-called semilattice graded weak Hopf algebra from a family of Hopf algebras. An example of semilattice graded weak Hopf algebra is just Clifford moniod algebra. Then, in Section 1.2, we give some properties and characterizations of this kind of weak Hopf algebra.The Maschke's Theorem of group algebras is well known, and [Mo2] gives us a version of Maschke's and Dual Maschke's Theorem of finite-dimensional Hopf algebras. At the side of the semigroup algebras, [CP] gives a similar result of inverse semigroup algebra. Since weak Hopf algebra is a generalization of monoid algebra, it is easy to ask what about the weak Hopf algebra? Semilattice graded weak Hopf algebra is a special case of weak Hopf algebra and a generalization of Hopf algebras, in Section 1.3, we will give results and applications to the Maschke's and Dual Maschke's Theorem of semilattice graded weak Hopf algebra.And, in Section 2.1, based on [L9], a regular solution of Yang-Baxter Equation and its decomposition can be obtained. Moreover, similar to the corresponding results in [L8], we will finish the decomposition and semi-simplicity of G-quantum doubles of semilattice graded weak Hopf algebras under the condition of commutativity.Although the quantum double of a finite Clifford monoid is indeed a generalization of the quantum double of a finite group, the quantum doubles in [L2] can not usually be regarded as...