2,891 research outputs found
Weak Projections onto a Braided Hopf Algebra
We show that, under some mild conditions, a bialgebra in an abelian and
coabelian braided monoidal category has a weak projection onto a formally
smooth (as a coalgebra) sub-bialgebra with antipode; see Theorem 1.12. In the
second part of the paper we prove that bialgebras with weak projections are
cross product bialgebras; see Theorem 2.12. In the particular case when the
bialgebra is cocommutative and a certain cocycle associated to the weak
projection is trivial we prove that is a double cross product, or biproduct
in Madjid's terminology. The last result is based on a universal property of
double cross products which, by Theorem 2.15, works in braided monoidal
categories. We also investigate the situation when the right action of the
associated matched pair is trivial
Cotensor Coalgebras in Monoidal Categories
We introduce the concept of cotensor coalgebra for a given bicomodule over a
coalgebra in an abelian monoidal category. Under some further conditions we
show that such a cotensor coalgebra exists and satisfies a meaningful universal
property. We prove that this coalgebra is formally smooth whenever the comodule
is relative injective and the coalgebra itself is formally smooth
The Heyneman-Radford Theorem for Monoidal Categories
We prove Heyneman-Radford Theorem in the framework of Monoidal Categories
Braided Bialgebras of Type One
Braided bialgebras of type one in abelian braided monoidal categories are
characterized as braided graded bialgebras which are strongly
-graded both as an algebra and as a coalgebra
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