122 research outputs found
Cocycle deformations for liftings of quantum linear spaces
Let be a Hopf algebra over a field of characteristic 0 and suppose
there is a coalgebra projection from to a sub-Hopf algebra that
splits the inclusion. If the projection is -bilinear, then is isomorphic
to a biproduct R #_{\xi}H where is called a pre-bialgebra with
cocycle in the category . The cocycle maps to . Examples of this situation include the liftings of pointed
Hopf algebras with abelian group of points as classified by
Andruskiewitsch and Schneider [AS1]. One asks when such an can be twisted
by a cocycle to obtain a Radford biproduct. By
results of Masuoka [Ma1, Ma2], and Gr\"{u}nenfelder and Mastnak [GM], this can
always be done for the pointed liftings mentioned above.
In a previous paper [ABM1], we showed that a natural candidate for a twisting
cocycle is {} where is a total
integral for and is as above. We also computed the twisting cocycle
explicitly for liftings of a quantum linear plane and found some examples where
the twisting cocycle we computed was different from {}. In
this note we show that in many cases this cocycle is exactly
and give some further examples where this is not the case. As well we extend
the cocycle computation to quantum linear spaces; there is no restriction on
the dimension
Quasi-bialgebra Structures and Torsion-free Abelian Groups
We describe all the quasi-bialgebra structures of a group algebra over a
torsion-free abelian group. They all come out to be triangular in a unique way.
Moreover, up to an isomorphism, these quasi-bialgebra structures produce only
one (braided) monoidal structure on the category of their representations.
Applying these results to the algebra of Laurent polynomials, we recover two
braided monoidal categories introduced in \cite{CG} by S. Caenepeel and I.
Goyvaerts in connection with Hom-structures (Lie algebras, algebras,
coalgebras, Hopf algebras)
Rota-Baxter operators on BiHom-associative algebras and related structures
The purpose of this paper is to study Rota-Baxter operators for
BiHom-associative algebras. Moreover, we introduce and discuss the properties
of the notions of BiHom-(tri)dendriform algebra, BiHom-Zinbiel algebra and
BiHom-quadri-algebra. We construct the free Rota-Baxter BiHom-associative
algebra and present some observations about categories and functors related to
Rota-Baxter structures.Comment: 27 pages. The title has been changed, the part about BiHom-Lie
algebras has been removed and will appear as a separate article. In v3 we
added some things about BiHom-Zinbiel algebra
BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras
A BiHom-associative algebra is a (nonassociative) algebra endowed with
two commuting multiplicative linear maps
such that , for all . This concept
arose in the study of algebras in so-called group Hom-categories. In this
paper, we introduce as well BiHom-Lie algebras (also by using the categorical
approach) and BiHom-bialgebras. We discuss these new structures by presenting
some basic properties and constructions (representations, twisted tensor
products, smash products etc)
BiHom-pre-Lie algebras, BiHom-Leibniz algebras and Rota-Baxter operators on BiHom-Lie algebras
We investigate some properties of Rota-Baxter operators on BiHom-Lie
algebras. Along the way, we introduce BiHom analogues of pre-Lie and Leibniz
algebras.Comment: 18 pages. The content of this paper was part of the first version of
arXiv:math.QA/1703.07275. In v2 we added a short section about BiHom-PostLie
algebra
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