3,398 research outputs found
Quasi-bialgebra Structures and Torsion-free Abelian Groups
We describe all the quasi-bialgebra structures of a group algebra over a
torsion-free abelian group. They all come out to be triangular in a unique way.
Moreover, up to an isomorphism, these quasi-bialgebra structures produce only
one (braided) monoidal structure on the category of their representations.
Applying these results to the algebra of Laurent polynomials, we recover two
braided monoidal categories introduced in \cite{CG} by S. Caenepeel and I.
Goyvaerts in connection with Hom-structures (Lie algebras, algebras,
coalgebras, Hopf algebras)
Squared Hopf algebras and reconstruction theorems
Given an abelian k-linear rigid monoidal category V, where k is a perfect
field, we define squared coalgebras as objects of cocompleted V tensor V
(Deligne's tensor product of categories) equipped with the appropriate notion
of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are
defined without use of braiding.
If V is the category of k-vector spaces, squared (co)algebras coincide with
conventional ones. If V is braided, a braided Hopf algebra can be obtained from
a squared one.
Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras
in V and corresponding fibre functors to V (which is not the case with other
definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to
the problem of defining quantum groups in braided categories.Comment: Latex2e, 31 pages, to appear in the Proceedings of Banach Center
Minisemester on Quantum Groups, November 199
Simplicial presheaves of coalgebras
The category of simplicial R-coalgebras over a presheaf of commutative unital
rings on a small Grothendieck site is endowed with a left proper, simplicial,
cofibrantly generated model category structure where the weak equivalences are
the local weak equivalences of the underlying simplicial presheaves. This model
category is naturally linked to the R-local homotopy theory of simplicial
presheaves and the homotopy theory of simplicial R-modules by Quillen
adjunctions. We study the comparison with the R-local homotopy category of
simplicial presheaves in the special case where R is a presheaf of
algebraically closed (or perfect) fields. If R is a presheaf of algebraically
closed fields, we show that the R-local homotopy category of simplicial
presheaves embeds fully faithfully in the homotopy category of simplicial
R-coalgebras.Comment: 24 page
The homotopy theory of coalgebras over a comonad
Let K be a comonad on a model category M. We provide conditions under which
the associated category of K-coalgebras admits a model category structure such
that the forgetful functor to M creates both cofibrations and weak
equivalences.
We provide concrete examples that satisfy our conditions and are relevant in
descent theory and in the theory of Hopf-Galois extensions. These examples are
specific instances of the following categories of comodules over a coring. For
any semihereditary commutative ring R, let A be a dg R-algebra that is
homologically simply connected. Let V be an A-coring that is semifree as a left
A-module on a degreewise R-free, homologically simply connected graded module
of finite type. We show that there is a model category structure on the
category of right A-modules satisfying the conditions of our existence theorem
with respect to the comonad given by tensoring over A with V and conclude that
the category of V-comodules in the category of right A-modules admits a model
category structure of the desired type. Finally, under extra conditions on R,
A, and V, we describe fibrant replacements in this category of comodules in
terms of a generalized cobar construction.Comment: 34 pages, minor corrections. To appear in the Proceedings of the
London Mathematical Societ
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