Braided Bialgebras of Type One: Applications

The main purpose of this paper is to study some relevant applications of the results on bialgebras of type one obtained in [AM1] in the framework of abelian braided monoidal categories. Let H be a braided bialgebra in a cocomplete and complete abelian braided monoidal category (M, c) satisfying AB5. Assume that the tensor product commutes with direct sums and is twosided exact. Let M be in HM H H . Let T = TH(M) be the relative tensor algebra and let T c = T c H(M) be the relative cotensor coalgebra as introduced in [AMS1]. Then both T and T c have a natural structure of graded braided bialgebra and the natural algebra morphism from T to T , which coincide with the canonical injections on H and M , is a graded bialgebra homomorphism. Thus its image is a graded braided bialgebra which is denoted by H [M ] and called, accordingly to [Ni], the braided bialgebra of type one associated to H and M . In order to characterize braided bialgebras of type one, we developed in [AM1] some useful properties of graded algebras and coalgebras. We introduced the definition of strongly N-graded algebras and coalgebras which is inspired in the second case to [NT]. A graded algebra A = ⊕n∈NAn in M is defined to be a strongly N-graded algebra when Ai+j = Ai ·A Aj for every i, j ∈ N. Dually, given a graded coalgebra (C = ⊕n∈NCn,∆, ε) in M, we can write ∆|Cn as the sum of unique components ∆i,j : Ci+j → Ci⊗Cj where i+ j = n. The coalgebra C is defined to be a strongly N-graded coalgebra when ∆i,j : Ci+j → Ci ⊗ Cj is a monomorphism for every i, j ∈ N. In this paper, by applying the results in [AM1], we prove that the associated graded coalgebra grCE := ⊕n∈N CE CE for a given subcoalgebra C of a coalgebra E in M, is a strongly N-graded coalgebra (see Theorem 2.10) and hence it can be characterized accordingly to [AM1] as in Theorem 2.12.