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Cocycle condition for multi-pullbacks of algebras

Piotr M. Hajac, Bartosz Zieliński (2012)

Banach Center Publications

Take finitely many topological spaces and for each pair of these spaces choose a pair of corresponding closed subspaces that are identified by a homeomorphism. We note that this gluing procedure does not guarantee that the building pieces, or the gluings of some pieces, are embedded in the space obtained by putting together all given ingredients. Dually, we show that a certain sufficient condition, called the cocycle condition, is also necessary to guarantee sheaf-like properties of surjective multi-pullbacks...

Differential calculus on almost commutative algebras and applications to the quantum hyperplane

Cătălin Ciupală (2005)

Archivum Mathematicum

In this paper we introduce a new class of differential graded algebras named DG ρ -algebras and present Lie operations on this kind of algebras. We give two examples: the algebra of forms and the algebra of noncommutative differential forms of a  ρ -algebra. Then we introduce linear connections on a  ρ -bimodule M over a  ρ -algebra  A and extend these connections to the space of forms from A to M . We apply these notions to the quantum hyperplane.

Homotopy theory of Hopf Galois extensions

Christian Kassel, Hans-Jürgen Schneider (2005)

Annales de l'institut Fourier

We introduce the concept of homotopy equivalence for Hopf Galois extensions and make a systematic study of it. As an application we determine all H -Galois extensions up to homotopy equivalence in the case when H is a Drinfeld-Jimbo quantum group.

Quantization of pencils with a gl-type Poisson center and braided geometry

Dimitri Gurevich, Pavel Saponov (2011)

Banach Center Publications

We consider Poisson pencils, each generated by a linear Poisson-Lie bracket and a quadratic Poisson bracket corresponding to a so-called Reflection Equation Algebra. We show that any bracket from such a Poisson pencil (and consequently, the whole pencil) can be restricted to any generic leaf of the Poisson-Lie bracket. We realize a quantization of these Poisson pencils (restricted or not) in the framework of braided affine geometry. Also, we introduce super-analogs of all these Poisson pencils and...

Shuffle bialgebras

María Ronco (2011)

Annales de l’institut Fourier

The goal of our work is to study the spaces of primitive elements of some combinatorial Hopf algebras, whose underlying vector spaces admit linear basis labelled by subsets of the set of maps between finite sets. In order to deal with these objects we introduce the notion of shuffle algebras, which are coloured algebras where composition is not always defined. We define bialgebras in this framework and compute the subpaces of primitive elements associated to them. These spaces of primitive elements...

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