Hopf代数的若干弱结构

研究了Hopf代数的一些弱概念及它们之间的关系，性质和特性，并刻画其上的模或余模结构，首先引入Hopf代数的一些弱化结构并讨论其关系，然后用某些弱Hopf代数的弱对极构造正则半群，另一方面，由可逆半群建构出一个弱Hopf代数，给出了余交换点双代数成为左/右Hopf代数的一些等价条件，利用不可约分支和类群元集么半群，在双代数（弱Hopf代数）中构造出一些子双代数（子弱Hopf代数），进一步，一些双代数被证明作为子双代数是不可约分支的补带/半格之和。当类群元么半群是Clifford么半群时，由一些不可约分支之和构造出一个左拟模双代数，最后给出的一些结果体现了弱时极在弱Hopf代数上的模/余模结构中的作用。

Some weaker structures of Hopf algebras

Some weaker concepts of Hopf algebras and their characterizations, relationships and the structures of modules and co-modules over them were studied. First, the concepts of weaker structures of Hopf algebras were introduced and their relationships were discussed. Then, some regular semigroups were constructed by using of weak antipodes of some weak Hopf algebras. Then a weak Hopf algebra was built from an inverse semigroup. Some sufficient and necessary conditions are given under which a cocommutative pointed bialgebra becomes a left/right Hopf algebra. By using irreducible components and group-like monoids, some sub-bialgebras and sub-weak Hopf algebras were constructed from a bialgebra or weak Hopf algebra. Moreover, it was shown that for a bialgebra, when the idempotent submonoid of the group-like monoid is a band (semi-lattice), there exists a sub-bialgebra which is a supplemental band (semi-lattice) sum of some sub-algebras without identities. When the group-like monoid was a Clifford monoid, a left quasi-module-bialgebra was built from the sum of some irreducible comonents. Last, the role of weak antipodes in the structures of modules/ comodules over a weak Hopf algebra is presented.