12 research outputs found
Doi-Koppinen modules for quantum groupoids
A definition of a Doi-Koppinen datum over a noncommutative algebra is
proposed. The idea is to replace a bialgebra in a standard Doi-Koppinen datum
with a bialgebroid. The corresponding category of Doi-Koppinen modules over a
noncommutative algebra is introduced. A weak Doi-Koppinen datum and module of
[G. Bohm. Comm. Algebra, 28:4687--4698, 2000] are shown to be examples of a
Doi-Koppinen datum and module over an algebra. A coring associated to a
Doi-Koppinen datum over an algebra is constructed and various properties of
induction and forgetful functors for Doi-Koppinen modules over an algebra are
deduced from the properies of corresponding functors in the category of
comodules of a coring.Comment: 14 pages, LaTeX; final version to appear in J. Pure Appl. Algebr
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
Serre Theorem for involutory Hopf algebras
We call a monoidal category a Serre category if for any ,
such that C\ot D is semisimple, and are
semisimple objects in . Let be an involutory Hopf algebra,
, two -(co)modules such that is (co)semisimple as a
-(co)module. If (resp. ) is a finitely generated projective
-module with invertible Hattory-Stallings rank in then (resp. )
is (co)semisimple as a -(co)module. In particular, the full subcategory of
all finite dimensional modules, comodules or Yetter-Drinfel'd modules over
the dimension of which is invertible in are Serre categories.Comment: a new version: 8 page
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