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Claudia Menini

Publications and source records attributed to Claudia Menini.

26 records · Page 2Linked to original sources
arXiv2013

Adjunctions and Braided Objects

In this paper we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behaviour of these functors in the case when the base category is braided is also considered.

math.CT↗
arXiv2013

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).

math.QA↗
arXiv2011

Gauge deformations for Hopf algebras with the dual Chevalley property

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 $ζ$ on $A$ such that $A^ζ\cong Q#H$ where $A^ζ$ is the dual quasi-bialgebra obtained from $A$ by twisting its multiplication by $ζ$, $Q$ 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 $H$.

math.QA↗
arXiv2011

Cocycle deformations for liftings of quantum linear spaces

Let $A$ be a Hopf algebra over a field $K$ of characteristic 0 and suppose there is a coalgebra projection $π$ 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 $R #_ξH$ where $(R,ξ)$ is called a pre-bialgebra with cocycle in the category $_{H}^{H}\mathcal{YD}$. The cocycle $ξ$ maps $R \otimes R$ to $H$. Examples of this situation include the liftings of pointed Hopf algebras with abelian group of points $Γ$ as classified by Andruskiewitsch and Schneider [AS1]. One asks when such an $A$ can be twisted by a cocycle $γ:A\otimes A\rightarrow K$ to obtain a Radford biproduct. By results of Masuoka [Ma1, Ma2], and Grünenfelder 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 {$λ\circ ξ$} where $λ\in H^{\ast}$ is a total integral for $H$ and $ξ$ 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 {$λ\circ ξ$}. In this note we show that in many cases this cocycle is exactly $λ\circξ$ 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.

math.QA↗
arXiv2010

Cocycle deformations for Hopf algebras with a coalgebra projection

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 $π$ from $A$ to $H$ which splits the inclusion. Then $A \cong R \#_ξH$ where $R$ is the pre-bialgebra of coinvariants. In this paper we study the deformations of $A$ by an $H$-bilinear cocycle. If $γ$ is a cocycle for $A$, then $γ$ can be restricted to a cocycle $γ_R$ for $R$, and $A^γ\cong R^{γ_R} \#_{ξ_γ} H$. As examples, we consider liftings of $\mathcal{B}(V) \# K[Γ]$ where $Γ$ 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.

math.QA↗
arXiv2008

Pre-torsors and Galois comodules over mixed distributive laws

We study comodule functors for comonads arising from mixed distributive laws. Their Galois property is reformulated in terms of a (so-called) regular arrow in Street's bicategory of comonads. Between categories possessing equalizers, we introduce the notion of a regular adjunction. An equivalence is proven between the category of pre-torsors over two regular adjunctions $(N_A,R_A)$ and $(N_B,R_B)$ on one hand, and the category of regular comonad arrows $(R_A,ξ)$ from some equalizer preserving comonad ${\mathbb C}$ to $N_BR_B$ on the other. This generalizes a known relationship between pre-torsors over equal commutative rings and Galois objects of coalgebras.Developing a bi-Galois theory of comonads, we show that a pre-torsor over regular adjunctions determines also a second (equalizer preserving) comonad ${\mathbb D}$ and a co-regular comonad arrow from ${\mathbb D}$ to $N_A R_A$, such that the comodule categories of ${\mathbb C}$ and ${\mathbb D}$ are equivalent.

math.RA↗
arXiv2008

A Schneider type theorem for Hopf algebroids

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.

math.QA↗