14 research outputs found
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
The Classification of All Crossed Products
Using the computational approach introduced in [Agore A.L., Bontea C.G.,
Militaru G., J. Algebra Appl. 12 (2013), 1250227, 24 pages, arXiv:1207.0411] we
classify all coalgebra split extensions of by , where is
the cyclic group of order and is Sweedler's -dimensional Hopf
algebra. Equivalently, we classify all crossed products of Hopf algebras by explicitly computing two classifying objects: the cohomological
'group' and
the set of types of isomorphisms of all crossed products .
More precisely, all crossed products are described by
generators and relations and classified: they are -dimensional quantum
groups , parameterized by the set of all pairs consisting of an arbitrary unitary map and an -th root
of . As an application, the group of Hopf algebra
automorphisms of is explicitly described
Crossed Product of Cyclic Groups
All crossed products of two cyclic groups are explicitly described using
generators and relations. A necessary and sufficient condition for an extension
of a group by a group to be a cyclic group is given.Comment: To appear in Czechoslovak Mathematical Journa
Schreier type theorems for bicrossed products
We prove that the bicrossed product of two groups is a quotient of the
pushout of two semidirect products. A matched pair of groups is deformed using a combinatorial datum consisting of
an automorphism of , a permutation of the set and a
transition map in order to obtain a new matched pair such that there exist an -invariant
isomorphism of groups . Moreover, if we fix the group and the automorphism
\sigma \in \Aut(H) then any -invariant isomorphism between two
arbitrary bicrossed product of groups is obtained in a unique way by the above
deformation method. As applications two Schreier type classification theorems
for bicrossed product of groups are given.Comment: 21 pages, final version to appear in Central European J. Mat
-universal Hopf algebras (co)acting on -algebras
We develop a theory which unifies the universal (co)acting bi/Hopf algebras
as studied by Sweedler, Manin and Tambara with the recently introduced
\cite{AGV1} bi/Hopf-algebras that are universal among all support equivalent
(co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a
family of linear maps between tensor powers of , called -algebras.
This allows us to treat algebras, coalgebras, braided vector spaces and many
other structures in a unified way. We study -universal measuring coalgebras
and -universal comeasuring algebras between -algebras and ,
relative to a fixed subspace of \Vect(A,B). By considering the case
, we derive the notion of a -universal (co)acting bialgebra (and Hopf
algebra) for a given algebra . In particular, this leads to a refinement of
the existence conditions for the Manin--Tambara universal coacting bi/Hopf
algebras. We establish an isomorphism between the -universal acting bi/Hopf
algebra and the finite dual of the -universal coacting bi/Hopf algebra under
certain conditions on in terms of the finite topology on \End_F(A).Comment: To appear in Commun. Contemp. Mat