860 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
Small Bialgebras with a Projection: Applications
In this paper we continue the investigation started in [A.M.St.-Small],
dealing with bialgebras with an -bilinear coalgebra projection over an
arbitrary subbialgebra with antipode. These bialgebras can be described as
deformed bosonizations R#_{\xi} H of a pre-bialgebra by with a
cocycle . Here we describe the behavior of in the case when is
f.d. and thin i.e. it is connected with one dimensional space of primitive
elements. This is used to analyze the arithmetic properties of . Meaningful
results are obtained when is cosemisimple. By means of Ore extension
construction, we provide some examples of atypical situations (e.g. the
multiplication of is not -colinear or is non-trivial)
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
Categories of comodules and chain complexes of modules
Let \lL(A) denote the coendomorphism left -bialgebroid associated to a
left finitely generated and projective extension of rings with
identities. We show that the category of left comodules over an epimorphic
image of \lL(A) is equivalent to the category of chain complexes of left
-modules. This equivalence is monoidal whenever is commutative and
is an -algebra. This is a generalization, using entirely new tools, of
results by B. Pareigis and D. Tambara for chain complexes of vector spaces over
fields. Our approach relies heavily on the non commutative theory of Tannaka
reconstruction, and the generalized faithfully flat descent for small additive
categories, or rings with enough orthogonal idempotents.Comment: The title has been changed, the first part is removed and the
construction of the coendomorphim bialgebroid is now freely used in the
statement of the main Theorem
The Heyneman-Radford Theorem for Monoidal Categories
We prove Heyneman-Radford Theorem in the framework of Monoidal Categories
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