arXiv subjects

D. Bulacu

Publications and source records attributed to D. Bulacu.

14 recordsLinked to original sources

arXiv2018

Frobenius and separable functors for the category of entwined modules over cowreaths, I: General theory

Entwined modules over cowreaths in a monoidal category are introduced. They can be identified to coalgebras in an appropriate monoidal category. It is investigated when such coalgebras are Frobenius (resp. separable), and when the forgetful functor from entwined modules to representations of the underlying algebra is Frobenius (resp. separable). These properties are equivalent when the unit object of the category is a $\otimes$-generator.

math.CT↗

arXiv2013

Monoidal ring and coring structures obtained from wreaths and cowreaths

Let $A$ be an algebra in a monoidal category $\Cc$, and let $X$ be an object in $\Cc$. We study $A$-(co)ring structures on the left $A$-module $A\ot X$. These correspond to (co)algebra structures in $EM(\Cc)(A)$, the Eilenberg-Moore category associated to $\Cc$ and $A$. The ring structures are in bijective correspondence to wreaths in $\Cc$, and their category of representations is the category of representations over the induced wreath product. The coring structures are in bijective correspondence to cowreaths in $\Cc$, and their category of corepresentations is the category of generalized entwined modules. We present several examples coming from (co)actions of Hopf algebras and their generalizations. Various notions of smash products that have appeared in the literature appear as special cases of our construction.

math.RA↗

arXiv2011

On integrals and cointegrals for quasi-Hopf algebras

Using the machinery provided by a Frobenius algebra we show how the antipode of a quasi-Hopf algebra $H$ carries out left or right cointegrals for $H$. These formulas will allow us to find out the explicit form of an integral and a cointegral for the quantum double $D(H)$ of $H$ in terms of those of $H$, and so to answer to a conjecture of Hausser and Nill raised at the end of the nineties.

math.QA↗

arXiv2011

On cross product Hopf algebras

Let $A$ and $B$ be algebras and coalgebras in a braided monoidal category $\Cc$, and suppose that we have a cross product algebra and a cross coproduct coalgebra structure on $A\ot B$. We present necessary and sufficient conditions for $A\ot B$ to be a bialgebra, and sufficient conditions for $A\ot B$ to be a Hopf algebra. We discuss when such a cross product Hopf algebra is a double cross (co)product, a biproduct, or, more generally, a smash (co)product Hopf algebra. In each of these cases, we provide an explicit description of the associated Hopf algebra projection.

math.QA↗

arXiv2010

A monoidal structure on the category of relative Hopf modules

Let $B$ be a bialgebra, and $A$ a left $B$-comodule algebra in a braided monoidal category $\Cc$, and assume that $A$ is also a coalgebra, with a not-necessarily associative or unital left $B$-action. Then we can define a right $A$-action on the tensor product of two relative Hopf modules, and this defines a monoidal structure on the category of relative Hopf modules if and only if $A$ is a bialgebra in the category of left Yetter-Drinfeld modules over $B$. Some examples are given.

math.CT↗

arXiv2010

Algebras graded by discrete Doi-Hopf data and the Drinfeld double of a Hopf group-coalgebra

We study Doi-Hopf data and Doi-Hopf modules for Hopf group-coalgebras. We introduce modules graded by a discrete Doi-Hopf datum; to a Doi-Hopf datum over a Hopf group coalgebra, we associate an algebra graded by the underlying discrete Doi-Hopf datum, using a smash product type construction. The category of Doi-Hopf modules is then isomorphic to the category of graded modules over this algebra. This is applied to the category of Yetter-Drinfeld modules over a Hopf group coalgebra, leading to the construction of the Drinfeld double. It is shown that this Drinfeld double is a quasitriangular graded Hopf algebra.

math.RA↗

arXiv2009

The braided monoidal structures on the category of vector spaces graded by the Klein group

Let $k$ be a field, $k^*=k\setminus\{0\}$ and $C_2$ the cyclic group of order 2. In this note we compute all the braided monoidal structures on the category of $k$-vector spaces graded by the Klein group $C_2\times C_2$. Actually, for the monoidal structures we will compute the explicit form of the 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$, while for the braided monoidal structures we will compute the explicit form of the abelian 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras, out of the vector space $k[C_2\times C_2]$.

math.QA↗

arXiv2008

Involutory quasi-Hopf algebras

We introduce and investigate the basic properties of an involutory (dual) quasi-Hopf algebra. We also study the representations of an involutory quasi-Hopf algebra and prove that an involutory dual quasi-Hopf algebra with non-zero integral is cosemisimple.

math.QA↗

arXiv2004

Doi-Hopf modules and Yetter-Drinfeld modules for quasi-Hopf algebras

For a quasi-Hopf algebra $H$, a left $H$-comodule algebra $\mf{B}$ and a right $H$-module coalgebra $C$ we will characterize the category of Doi-Hopf modules ${}^C{\cal M}(H)_{\mf{B}}$ in terms of modules. We will also show that for an $H$-bicomodule algebra $\mb{A}$ and an $H$-bimodule coalgebra $C$ the category of generalized Yetter-Drinfeld modules ${}_{\mb{A}}{\cal YD}(H)^C$ is isomorphic to a certain category of Doi-Hopf modules. Using this isomorphism we will transport the properties from the category of Doi-Hopf modules to the category of generalized Yetter-Drinfeld modules.

math.QA↗

arXiv2004

Yetter-Drinfeld categories for quasi-Hopf algebras

We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra $H$ are isomorphic. We prove also that the category $\yd^{\rm fd}$ of finite dimensional left Yetter-Drinfeld modules is rigid and then we compute explicitly the canonical isomorphisms in $\yd^{\rm fd}$. Finally, we show that certain duals of $H_0$, the braided Hopf algebra introduced in \cite{bn,bpv}, are isomorphic as braided Hopf algebras if $H$ is a finite dimensional triangular quasi-Hopf algebra.

math.QA↗

arXiv2003

More properties of Yetter-Drinfeld modules over quasi-Hopf algebras

We generalize various properties of Yetter-Drinfeld modules over Hopf algebras to quasi-Hopf algebras. The dual of a finite dimensional Yetter-Drinfeld module is again a Yetter-Drinfeld module. The algebra $H_0$ in the category of Yetter-Drinfeld modules that can be obtained by modifying the multiplication in a proper way is quantum commutative. We give a Structure Theorem for Hopf modules in the category of Yetter-Drinfeld modules, and deduce the existence and uniqueness of integrals from it.

math.QA↗

arXiv2002

Two-sided (two-cosided) Hopf modules and Doi-Hopf modules for quasi-Hopf algebras

Let $H$ be a finite dimensional quasi-Hopf algebra over a field $k$ and ${\mathfrak A}$ a right $H$-comodule algebra in the sense of Hausser and Nill. We first show that on the $k$-vector space ${\mathfrak A}\ot H^*$ we can define an algebra structure, denoted by ${\mathfrak A}\ovsm H^*$, in the monoidal category of left $H$-modules (i.e. ${\mathfrak A}\ovsm H^*$ is an $H$-module algebra. Then we will prove that the category of two-sided $({\mathfrak A}, H)$-bimodules $\hba $ is isomorphic to the category of relative $({\mathfrak A}\ovsm H^*, H^*)$-Hopf modules, as introduced in by Hausser and Nill. In the particular case where ${\mathfrak A}=H$, we will obtain a result announced by Nill. We will also introduce the categories of Doi-Hopf modules and two-sided two-cosided Hopf modules and we will show that they are in certain situations isomorphic to module categories.

math.QA↗

arXiv2002

Integrals for (dual) quasi-Hopf algebras. Applications

A classical result in the theory of Hopf algebras concerns the uniqueness and existence of integrals: for an arbitrary Hopf algebra, the integral space has dimension $\leq 1$, and for a finite dimensional Hopf algebra, this dimension is exaclty one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode follows from the other axioms of a finite dimensional quasi-Hopf algebra. We give a new version of the Fundamental Theorem for quasi-Hopf algebras. We show that a dual quasi-Hopf algebra is co-Frobenius if and only if it has a non-zero integral. In this case, the space of left or right integrals has dimension one.

math.QA↗

arXiv2001

The quantum double for quasitriangular quasi-Hopf algebras

Let $D(H)$ be the quantum double associated to a finite dimensional quasi-Hopf algebra $H$. In this note, we first generalize a result of Majid, stating that a finite dimensional Hopf algebra $H$ is quasitriangular if and only if there is a projection of the quantum double $D(H)$ onto $H$ covering the natural inclusion.We then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct.

math.QA↗