by Iovanov, Miodrag Cristian

It is well known that the torsion part of any finitely generated module over the formal power series ring K[[X]] is a direct summand. In fact, K[[X]] is an algebra dual to the divided power coalgebra over K and the torsion part of any K[[X]]-module actually identifies with the rational part of that module. More generally, for a certain general enough class of coalgebras—those having only finite dimensional subcomodules—we see that the above phenomenon is preserved: the set of torsion elements of any C
*-module is exactly the rational submodule. With

by Böhm, Gabriella; Ştefan, Dragoş

In a recent paper (Böhm and Stefan, Commun Math Phys 282:239–286, 2008), we gave a general construction of a para-cocyclic structure on a cosimplex, associated to a so called admissible septuple—consisting of two categories, three functors and two natural transformations, subject to compatibility relations. The main examples of such admissible septuples were induced by algebra homomorphisms. In this note we provide more general examples coming from appropriate (‘locally braided’) morphisms of monads.

DOI: 10.1007/s10468-009-9160-7
Online Date: 3/26/2009
Print publication date: 10/1/2009
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by Beggs, E. J.; Majid, S.

We introduce the notion of ‘bar category’ by which we mean a monoidal category equipped with additional structure formalising the notion of complex conjugation. Examples of our theory include bimodules over a *-algebra, modules over a conventional *-Hopf algebra and modules over a more general object which we call a ‘quasi-*-Hopf algebra’ and for which examples include the standard quantum groups $u_q(\mathfrak{g})$ at q a root of unity (these are well-known not to be usual *-Hopf algebras). We also provide examples of strictly quasiassociative bar categories, including modules over ‘*-quasiHopf algebras’ and a construction based on finite subgroups H ⊂ G of a finite group. Inside a bar category one has natural notions of

by

DOI: 10.1007/s10468-009-9162-5
Online Date: 3/24/2009
Print publication date: 10/1/2009
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by Ardizzoni, A.; Menini, C.

In this paper we introduce the notions of connected, 0-connected and strictly graded coalgebra in the framework of an abelian monoidal category $ \mathcal{M} $ and we investigate the relations between these concepts. We recover several results, involving these notions, which are well known in the case when $ \mathcal{M} $ is the category of vector spaces over a field K. In particular we characterize when a 0-connected graded bialgebra is a bialgebra of type one.

DOI: 10.1007/s10468-009-9147-4
Online Date: 3/21/2009
Print publication date: 10/1/2009
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by Bican, Ladislav

At the beginning of this note the $\mathcal{G}$-covers, $\mathcal{G}$ being a hereditary class of modules, are characterized as that for which the homomorphisms into $\mathcal{G}$-precovers are injective as well as that for which the homomorphisms from $\mathcal{G}$-precovers are surjective. The next part studies the (pre)covers of (relatively) injective modules and some relations between the (relative) injectivity of modules and their (pre)covers.

DOI: 10.1007/s10468-009-9146-5
Online Date: 3/21/2009
Print publication date: 10/1/2009
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by Darpö, Erik

A rotation in a Euclidean space V is an orthogonal map δ ∈ O(V) which acts locally as a plane rotation with some fixed angle a(δ) ∈ [0,π]. We give a classification of all finite-dimensional representations of the real algebra $\mathbb{R}\left\langle X,Y\right\rangle$ that are given by rotations, up to orthogonal isomorphism.

DOI: 10.1007/s10468-009-9156-3
Online Date: 3/17/2009
Print publication date: 10/1/2009
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by Van den Bergh, Michel

This is a companion note to “Hochschild cohomology and Atiyah classes” by Damien Calaque and the author. Using elementary methods we compute the Kontsevich weight of a wheel with spokes pointing outward. The result is in terms of modified Bernoulli numbers. The same result had been obtained earlier by Torossian (unpublished) and also recently by Thomas Willwacher using more advanced methods.

DOI: 10.1007/s10468-009-9161-6
Online Date: 3/13/2009
Print publication date: 10/1/2009
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by Huang, Zhaoyong

Let R be a left Noetherian ring, S a right Noetherian ring and
R

U a generalized tilting module with S = End(