106 research outputs found
Local units versus local projectivity. Dualisations: Corings with local structure maps
We unify and generalize different notions of local units and local
projectivity. We investigate the connection between these properties by
constructing elementary algebras from locally projective modules. Dual versions
of these constructions are discussed, leading to corings with local
comultiplications, corings with local counits and rings with local
multiplications.Comment: 22 pages, including a correction to Proposition 1.
Lie monads and dualities
We study dualities between Lie algebras and Lie coalgebras, and their
respective (co)representations. To allow a study of dualities in an
infinite-dimensional setting, we introduce the notions of Lie monads and Lie
comonads, as special cases of YB-Lie algebras and YB-Lie coalgebras in additive
monoidal categories. We show that (strong) dualities between Lie algebras and
Lie coalgebras are closely related to (iso)morphisms between associated Lie
monads and Lie comonads. In the case of a duality between two Hopf algebras -in
the sense of Takeuchi- we recover a duality between a Lie algebra and a Lie
coalgebra -in the sense defined in this note- by computing the primitive and
the indecomposables elements, respectively.Comment: 27 pages, v2: some examples added and minor change
Morita theory of comodules over corings
By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita
context induces an equivalence between appropriate subcategories of the module
categories of the two rings in the Morita context. These are in fact categories
of firm modules for non-unital subrings. We apply this result to various Morita
contexts associated to a comodule of an -coring \cC. This allows
to extend (weak and strong) structure theorems in the literature, in particular
beyond the cases when any of the coring \cC or the comodule is
finitely generated and projective as an -module. That is, we obtain
relations between the category of \cC-comodules and the category of firm
modules for a firm ring , which is an ideal of the endomorphism algebra
^\cC(\Sigma). For a firmly projective comodule of a coseparable coring we
prove a strong structure theorem assuming only surjectivity of the canonical
map.Comment: LaTeX, 35 pages. v2: Minor changes including the title, examples
added in Section
Dual Constructions for Partial Actions of Hopf Algebras
The duality between partial actions (partial -module algebras) and
co-actions (partial -comodule algebras) of a Hopf algebra is fully
explored in this work. A connection between partial (co)actions and Hopf
algebroids is established under certain commutativity conditions. Moreover, we
continue this duality study, introducing also partial -module coalgebras and
their associated -rings, partial -comodule coalgebras and their
associated cosmash coproducts, as well as the mutual interrelations between
these structures.Comment: v3: strongly revised versio
Geometrically Partial actions
We introduce "geometric" partial comodules over coalgebras in monoidal
categories, as an alternative notion to the notion of partial action and
coaction of a Hopf algebra introduced by Caenepeel and Janssen. The name is
motivated by the fact that our new notion suits better if one wants to describe
phenomena of partial actions in algebraic geometry. Under mild conditions, the
category of geometric partial comodules is shown to be complete and cocomplete
and the category of partial comodules over a Hopf algebra is lax monoidal. We
develop a Hopf-Galois theory for geometric partial coactions to illustrate that
our new notion might be a useful additional tool in Hopf algebra theory.Comment: revised version; improved presentation; stronger version of
"fundamental theorem" for partial comodules. Version accepted for publication
in "Transactions of the American Mathematical Society". Updated reference
Globalization for geometric partial comodules
We discuss globalization for geometric partial comodules in a monoidal
category with pushouts and we provide a concrete procedure to construct it,
whenever it exists. The mild assumptions required by our approach make it
possible to apply it in a number of contexts of interests, recovering and
extending numerous ad hoc globalization constructions from the literature in
some cases and providing obstruction for globalization in some other cases.Comment: 18 pages. Major revision. Results and global presentation improved.
Comments are welcome
Equivalences of (co)module algebra structures over Hopf algebras
We introduce the notion of support equivalence for (co)module algebras (over
Hopf algebras), which generalizes in a natural way (weak) equivalence of
gradings. We show that for each equivalence class of (co)module algebra
structures on a given algebra A, there exists a unique universal Hopf algebra H
together with an H-(co)module structure on A such that any other equivalent
(co)module algebra structure on A factors through the action of H. We study
support equivalence and the universal Hopf algebras mentioned above for group
gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative
Hopf algebras. We show how the notion of support equivalence can be used to
reduce the classification problem of Hopf algebra (co)actions. We apply support
equivalence in the study of the asymptotic behaviour of codimensions of
H-identities and, in particular, to the analogue (formulated by Yu. A.
Bahturin) of Amitsur's conjecture, which was originally concerned with ordinary
polynomial identities. As an example we prove this analogue for all unital
H-module structures on the algebra of dual numbers.Comment: 35 pages; to appear in Journal of Noncommutative Geometr
A semi-abelian extension of a theorem by Takeuchi
We prove that the category of cocommutative Hopf algebras over a field is a
semi-abelian category. This result extends a previous special case of it, based
on the Milnor-Moore theorem, where the field was assumed to have zero
characteristic. Takeuchi's theorem asserting that the category of commutative
and cocommutative Hopf algebras over a field is abelian immediately follows
from this new observation. We also prove that the category of cocommutative
Hopf algebras over a field is action representable. We make some new
observations concerning the categorical commutator of normal Hopf subalgebras,
and this leads to the proof that two definitions of crossed modules of
cocommutative Hopf algebras are equivalent in this context.Comment: 27 page
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