Publications (11)0 Total impact
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ABSTRACT: 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).
02/2013;
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ABSTRACT: We show that the functor from bialgebras to vector spaces sending a bialgebra
to its subspace of primitives has monadic length at most 2.
03/2012;
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ABSTRACT: In this paper, we associate a dual quasi-bialgebra, called bosonization, to
every dual quasi-bialgebra $H$ and every bialgebra $R$ in the category of
Yetter-Drinfeld modules over $H$. Then, using the fundamental theorem, we
characterize as bosonizations the dual quasi-bialgebras with a projection onto
a dual quasi-bialgebra with a preantipode. As an application we investigate the
structure of the graded coalgebra $grA$ associated to a dual quasi-bialgebra
$A$ with the dual Chevalley property (e.g. $A$ is pointed).
11/2011;
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ABSTRACT: Let $A$ be a Hopf algebra over a field $K$ of characteristic zero such that
its coradical $H$ is a finite dimensional sub-Hopf algebra. Our main theorem
shows that there is a gauge transformation $\zeta $ on $A$ such that
$A^{\zeta}\cong Q#H$ where $A^\zeta$ is the dual quasi-bialgebra obtained from
$A$ by twisting its multiplication by $\zeta$, $Q$ is a connected dual
quasi-bialgebra in $^H_H\mathcal{YD}$ and $Q #H $ is a dual quasi-bialgebra
called the bosonization of $Q$ by $H$.
12/2010;
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ABSTRACT: It is known that a dual quasi-bialgebra with antipode $H$, i.e. a dual quasi-Hopf algebra, fulfils a fundamental theorem for right dual quasi-Hopf $H$-bicomodules. The converse in general is not true. We prove that, for a dual quasi-bialgebra $H$, the structure theorem amounts to the existence of a suitable map $S:H\rightarrow H$ that we call a preantipode of $H$.
12/2010;
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ABSTRACT: Let $A$ be a Hopf algebra over a field $K$ of characteristic 0 and suppose
there is a coalgebra projection $\pi$ from $A$ to a sub-Hopf algebra $H$ that
splits the inclusion. If the projection is $H$-bilinear, then $A$ is isomorphic
to a biproduct $R #_{\xi}H$ where $(R,\xi)$ is called a pre-bialgebra with
cocycle in the category $_{H}^{H}\mathcal{YD}$. The cocycle $\xi$ maps $R
\otimes R$ to $H$. Examples of this situation include the liftings of pointed
Hopf algebras with abelian group of points $\Gamma$ as classified by
Andruskiewitsch and Schneider [AS1]. One asks when such an $A$ can be twisted
by a cocycle $\gamma:A\otimes A\rightarrow K$ 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 {$\lambda \circ \xi$} where $\lambda\in H^{\ast}$ is a total
integral for $H$ and $\xi$ 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 {$\lambda \circ \xi$}. In
this note we show that in many cases this cocycle is exactly $\lambda\circ\xi$
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.
11/2010;
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Alessandro Ardizzoni
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ABSTRACT: We continue our investigation of the general notion of universal enveloping algebra that we introduced in [A. Ardizzoni, "A Milnor-Moore Type Theorem for Primitively Generated Braided Bialgebras", J. Algebra, to appear. (doi:10.1016/j.jalgebra.2010.07.031)]. Namely we study when such an algebra is of PBW type, meaning that a suitable PBW type theorem holds. We prove this is equivalent to require that the enveloping algebra is cosymmetric. We characterize braided bialgebra liftings of Nichols algebras as universal enveloping algebras of PBW type. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: As an application we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra.
08/2010;
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ABSTRACT: In this paper, the notion of universal enveloping algebra introduced in [A. Ardizzoni, \emph{A First Sight Towards Primitively Generated Connected Braided Bialgebras}, submitted. (arXiv:0805.3391v3)] is specialized to the case of braided vector spaces whose Nichols algebra is quadratic as an algebra. In this setting a classification of universal enveloping algebras for braided vector spaces of dimension not greater than 2 is handled. As an application, we investigate the structure of primitively generated connected braided bialgebras whose braided vector space of primitive elements forms a Nichols algebra which is quadratic algebra.
06/2009;
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ABSTRACT: Let $H$ be a Hopf algebra over a field $K$ of characteristic $0$ and let $A$ be a bialgebra or Hopf algebra such that $H$ is isomorphic to a sub-Hopf algebra of $A$ and there is an $H$-bilinear coalgebra projection $\pi$ from $A$ to $H$ which splits the inclusion. Then $A \cong R \#_\xi H$ where $R$ is the pre-bialgebra of coinvariants. In this paper we study the deformations of $A$ by an $H$-bilinear cocycle. If $\gamma$ is a cocycle for $A$, then $\gamma$ can be restricted to a cocycle $\gamma_R$ for $R$, and $A^\gamma \cong R^{\gamma_R} \#_{\xi_\gamma} H$. As examples, we consider liftings of $\mathcal{B}(V) \# K[\Gamma]$ where $\Gamma$ is a finite abelian group, $V$ is a quantum plane and $\mathcal{B}(V)$ is its Nichols algebra, and explicitly construct the cocycle which twists the Radford biproduct into the lifting. Comment: J. Algebra, to appear
06/2009;
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ABSTRACT: Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e.
algebra extensions B\subseteq A by H, are studied. Assuming that a lifted
canonical map is a split epimorphism of modules of the non-commutative base
algebra of H, relative injectivity of the H-comodule algebra A is related to
the Galois property of the extension B\subseteq A and also to the equivalence
of the category of relative Hopf modules to the category of B-modules. This
extends a classical theorem by H.-J. Schneider on Galois extensions by a Hopf
algebra. Our main tool is an observation that relative injectivity of a
comodule algebra is equivalent to relative separability of a forgetful functor,
a notion introduced and analysed hereby.
In the first version of this submission, we heavily used the statement that
two constituent bialgebroids in a Hopf algebroid possess isomorphic comodule
categories. This statement was based on \cite[Brz3,Theorem 2.6], whose proof
turned out to contain an unjustified step. In the revised version we return to
an earlier definition of a comodule of a Hopf algebroid, that distinguishes
between comodules of the two constituent bialgebroids, and modify the
statements and proofs in the paper accordingly.
12/2006;
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Alessandro Ardizzoni
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ABSTRACT: The natural problem we approach in the present paper is to show how the notion of formally smooth (co)algebra inside monoidal categories can substitute that of (co)separable (co)algebra in the study of splitting bialgebra homomorphisms. This is performed investigating the relation between formal smoothness and separability of certain functors and led to other results related to Hopf algebra theory. Between them we prove that the existence of $ad$-(co)invariant integrals for a Hopf algebra $H$ is equivalent to the separability of some forgetful functors. In the finite dimensional case, this is also equivalent to the separability of the Drinfeld Double $D(H)$ over $H$. Hopf algebras which are formally smooth as (co)algebras are characterized. We prove that given a bialgebra surjection $\pi :E\to H$ with nilpotent kernel such that $H$ is a Hopf algebra which is formally smooth as a $K$-algebra, then $\pi $ has a section which is a right $H$-colinear algebra homomorphism. Moreover, if $H$ is also endowed with an $ad$-invariant integral, then this section can be chosen to be $H$-bicolinear. We also deal with the dual case.
07/2004;
