Weak Hopf Group - Dual Mode And Structure Theorem

Weak Hopf algebra is defined by Bohm and Nill. As a generalization of Hopf algebra, it doesn’t need that the comultiplication keeps unit, and the counit is an algebra homomorphism, which makes properties of Hopf algebra have many weak fomality. V. G Turave introduced a new algebraic structure which was the Hopf π-coalgebra in2002. Hopf π-coalgebra was a generalization of Hopf algebra. Virelizier studied detailedly Hopf π-coalgebra. Meanwhile the notions of π-algebra and Hopf π-algebra were introduced and studied. Upon the backgroud above, in this paper we mainly talk about some properties of weak Hopf π-algebra H, duality and the structure theorem of weak Hopf π-H-module.The paper is organized as follows. We introduce first the basic notions about π-algebra, weak Hopf π-algebra, weak Hopf π-H-module and so on. Then, we give and prove some basic properties of weak Hopf π-algebra. One is that if H-({Hα, Δα,εα}α∈π, m, u, S) is a weak Hopf π-algebra, then the antipode S is a family of coalgebra anti-homomorphisms and a n-algebra anti-homomorphism. Secondly, if both weak Hopf π-algebra H and weak Hopf π-H-module M are locally finite dimension, we prove that the dual M*={Mα*}α∈π is a weak Hopf π-H*-comodule over weak Hopf π-coalgebra H*. Finally, we define a coinvariant subcomodule M1coH1of the weak Hopf π-H-module M, and show that M1coH1(?)H is a weak Hopf π-H-module. Then we prove the structure theorem of weak Hopf π-H-module, that is, if H is a weak Hopf π-algebra and M=({Mα}α∈π ζ,ρ) is a weak Hopf π-H-module, then as weak Hopf π-H-module M(?)M1coH1(?)H. As a corollary, let π={1}, we get the fundamental theorem of usual weak Hopf H-module.