The Cohomology Over Triangular Hopf Algebra

The main content of this paper is the cohomologies and extensions over triangular Hopf algebra. According to [l],we consider the cohomologies and extensions and then the relationships between them in the category of the modules of triangular Hopf algebra H.And the category is a symmetry monoidal category.In the first section , we introduce some knowledge we need in this article .They are the definitions of quasitriangular Hopf algebra.triangular Hopf algebra . the theorem of the property of their modules category , and the property of R.In the second section . according to the need of this article . we introduce some basic concepts in the category of H- modules. For example : H -module algebras. H- module coalgebras. H- module bialgebras. H- modules Hopf algeras and left .4-modules.left .4-module algebras for //-module algebra .4 in the category of H-modules and right B- comodules. right B-comodule algebras,right B- comodule coalgebras for H- module coalgebra B in the category of H- modules and then the categories for them accordingly . we prove some propoties.such as: 2.3 notations (2)the tensor product of commutative H-module algebras is also a commutative H-module algebra . And 2.6 propIn the third section , we introduce the cohomology of H-module bialgebra B in left B- module M in the category of H-modules . For this , we proved the following theroem firstly .3.1 propsition : B is a cocommutative H-module bialgebra,then there is an adjoint functor pair < T, U >, where: U : (H,B)Coalg HCoalg is a forgetful functor; T : HCoalg (H,B)Coalg is defined as following :We dicussed the left B -module structure of B which is and the m times coproduct formulation . Then in order to define the cohomology .according to the adjoint pair in 3.1, we defined semisimplicial complex and use (H.B)Reg(--) we have the cosimplicial complex accordingly. For the purpose of compute .according to 2. 6. we have :3.7The realization of complex We have complex and it's normalized subcomplex. which is isomorphic to . At the last of this section .we compute 2-cocycle and 2-cobounder concretely.In the forth section, we introduced extension of H- module algebras. At first, we give the definition of the extension. Then we give some important maps :We use present and proved some important equations about above maps and the extensions .4.2 theorem : Let E is an extension of H-module algebras, is the system from E , then we have :4.3 B is a commutative H-module bialgebra . M is a commutative H module algebra . Let .is a morphism of H-modules . V a, b e B m, n 6 M Let:then is an extension of H- module algebras . satsfy (iv}(v)(vi)in theorem 2.And then we gave the definition of a mophism of the extensions of H-module algebras and proved some relations.4.5 prop:is a morphism of exteneions of H- modulealgebras , are the systems from E1,E2 Let :then And the sufficiently and needly conditions which make a morphism of H-module to the morphism of H-module algebras' exteneions . At the last we dually wrote the definition of an extension of H-module coalgebras and the properties accordingly .In the last section ,we discussed the relationship between the extensions of H-module algebras and the cohomology . At first .we have the property of cotensor product of left, reght B- comodule in H.M.5.1 prop : B is a cocommutative H-module bialgebra . are left . right B-comodule algebra in ,if, there is a cotensor peoduct of B- comodule in then there is one and only one morphism : is the H-module structure of is the morphism of H-module algebras .Then we gave the definition of regular extension of H-module algebras, and a sufficiently and needly condition which make an extension in a regular one :5.3 theorem: M is a commutative H-module algebra, B is a cocommu-tative H-module Hopf algebra, the antipode is SB- Let : is right B- comodule algebra in HA4 , then the following equavelent : (i)there is a map a, , satisfy: is a regular extension of H-module alge...