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.
Let $H$ be a $\times$-bialgebra in the sense of Takeuchi. We show that if $H$ is $\times$-Hopf, and if $H$ fulfills the finiteness condition necessary to define its skew dual $H^\vee$, then the coopposite of the latter is $\times$-Hopf as well. If in addition the coopposite $\times$-bialgebra of $H$ is $\times$-Hopf, then the coopposite of the Drinfeld double of $H$ is $\times$-Hopf, as is the Drinfeld double itself, under an additional finiteness condition.
We provide a quiver setting for quasi-Hopf algebras, generalizing the Hopf quiver theory. As applications we obtain some general structure theorems, in particular the quasi-Hopf analogue of the Cartier theorem and the Cartier-Gabriel decomposition theorem.
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.
We find a new class of Hopf algebras, local quasitriangular Hopf algebras, which generalize quasitriangular Hopf algebras. Using these Hopf algebras, we obtain solutions of the Yang-Baxter equation in a systematic way. The category of modules with finite cycles over a local quasitriangular Hopf algebra is a braided tensor category.
We give a coring version for the duality theorem for actions and coactions of a finitely generated projective Hopf algebra. We also provide a coring analogue for a theorem of H.-J. Schneider, which generalizes and unifies the duality theorem for finite Hopf algebras and its refinements.
Let $H$ be an infinite-dimensional braided Hopf algebra and assume that the braiding is symmetric on $H$ and its quasi-dual $H^d$. We prove the Blattner-Montgomery duality theorem, namely we prove $$ (R # H)# H^{d} \cong R \otimes (H # H^{d}) \hbox {as algebras in braided tensor category} {\cal C}.$$ In particular, we present two duality theorems for infinite braided Hopf algebras in the Yetter-Drinfeld module category.
We extend to the co-Frobenius case a result of Drinfeld and Radford related to the fourth power of the antipode of a finite dimensional (co)quasitriangular Hopf algebra.
The Structure Theorem for Hopf modules states that if a bialgebra $H$ is a Hopf algebra (i.e. it is endowed with a so-called antipode) then every Hopf module $M$ is of the form ${M}^{\mathrm{co}{H}}\otimes H$, where ${M}^{\mathrm{co}{H}}$ denotes the space of coinvariant elements in $M$. Actually, it has been shown that this result characterizes Hopf algebras: $H$ is a Hopf algebra if and only if every Hopf module $M$ can be decomposed in such a way. The main aim of this paper is to extend this characterization to the framework of quasi-bialgebras by introducing the notion of preantipode and b…
This paper is devoted to the study of the quasitriangularity of Hopf algebras via Hopf quiver approaches. We give a combinatorial description of the Hopf quivers whose path coalgebras give rise to coquasitriangular Hopf algebras. With a help of the quiver setting, we study general coquasitriangular pointed Hopf algebras and obtain a complete classification of the finite-dimensional ones over an algebraically closed field of characteristic 0.
The finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra $H$ is studied. Such a Hopf algebra $H$ is an extension of an affine commutative Hopf algebra $A$ by a finite dimensional Hopf algebra $F$. The main theorem gives natural conditions under which $H^{\circ}$ decomposes as a crossed or smash product of $F^{\ast}$ by the finite dual of $A$. This decomposition is then further analysed using the Cartier- Gabriel-Kostant theorem to obtain component Hopf subalgebras of $H^{\circ}$ mapping onto the classical components of $A^{\circ}$. The detailed consequences for a number of f…
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.
The duality between partial actions (partial $H$-module algebras) and co-actions (partial $H$-comodule algebras) of a Hopf algebra $H$ is fully explored in this work. A connection between partial (co)actions and Hopf algebroids is established under certain commutativity conditions. Moreover, we continue this duality study, introducing also partial $H$-module coalgebras and their associated $C$-rings, partial $H$-comodule coalgebras and their associated cosmash coproducts, as well as the mutual interrelations between these structures.
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 $H$.
We study quasi-Hopf algebras and their subobjects over certain commutative rings from the point of view of Frobenius algebras. We introduce a type of Radford formula involving an anti-automorphism and the Nakayama automorphism of a Frobenius algebra, then view several results in quantum algebras from this vantage-point. In addition, separability and strong separability of quasi-Hopf algebras are studied as Frobenius algebras.
This paper is an attempt to construct a special kind of Hopf pairing $\langle-,-\rangle:H^\bullet\otimes H\rightarrow\Bbbk$. Specifically, $H^\bullet$ and $H$ should be both affine, noetherian and of the same GK-dimension. In addition, some properties of them would be dual to each other. We test the ideas in two steps for all the affine prime regular Hopf algebras $H$ of GK-dimension one: 1) We compute the finite duals $H^\circ$ of them, which are given by generators and relations; 2) the Hopf pairings desired are determined by choosing certain Hopf subalgebras $H^\bullet$ of $H^\circ$, where …
This paper studies the duals of some finite dimensional pointed Hopf algebras, with abelian group of grouplikes, over an algebraically closed field of characteristic 0, which are either Radford biproducts or else nontrivial liftings of a biproduct.
We explore special features of the pair (U^*, U_*) formed by the right and left dual over a (left) bialgebroid U in case the bialgebroid is, in particular, a left Hopf algebroid. It turns out that there exists a bialgebroid morphism S^* from one dual to another that extends the construction of the antipode on the dual of a Hopf algebra, and which is an isomorphism if U is both a left and right Hopf algebroid. This structure is derived from Phung's categorical equivalence between left and right comodules over U without the need of a (Hopf algebroid) antipode, a result which we review and extend…
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…
In this paper, we study the antipode of a finite-dimensional Hopf algebra $H$ with the dual Chevalley property and obtain an annihilation polynomial for its antipode $S$. The annihilation polynomial is determined by the exponent $N$ of the coradical and the Loewy length. In particular, the order of $S^2$ divides $N$ in characteristic $0$. Moreover, we get two characterizations of the quasi-exponent.
A basis B of a finite dimensional Hopf algebra H is said to be positive if all the structure constants of H relative to B are non-negative. A quasi-triangular structure $R\in H\otimes H$ is said to be positive with respect to B if it has non-negative coefficients in the basis $B \otimes B$ of $H\otimes H$. In our earlier work, we have classified all finite dimensional Hopf algebras with positive bases. In this paper, we classify positive quasi-triangular structures on such Hopf algebras. A consequence of this classification is a new way of constructing set-theoretical solutions of the Yang-Bax…
Let $q>2$ be a prime number, $d$ be an odd square-free natural number, and $n$ be a non-negative integer. We prove that a semisimple quasitriangular Hopf algebra of dimension $dq^n$ is solvable in the sense of Etingof, Nikshych and Ostrik. In particular, if $n\leq 3$ then it is either isomorphic to $k^G$ for some abelian group $G$, or twist equivalent to a Hopf algebra which fits into a cocentral abelian exact sequence.
We introduce partial (co)actions of a Hopf algebra $H$ on an algebra. To this end, we introduce first the notion of lax coring, generalizing Wisbauer's notion of weak coring. We also have the dual notion of lax ring. Several duality results are given, and we develop Galois theory for partial $H$-comodule algebras.
We introduce the notions of Hopf quasigroup and Hopf coquasigroup $H$ generalising the classical notion of an inverse property quasigroup $G$ expressed respectively as a quasigroup algebra $k G$ and an algebraic quasigroup $k[G]$. We prove basic results as for Hopf algebras, such as anti(co)multiplicativity of the antipode $S:H\to H$, that $S^2=\id$ if $H$ is commutative or cocommutative, and a theory of crossed (co)products. We also introduce the notion of a Moufang Hopf (co)quasigroup and show that the coordinate algebras $k[S^{2^n-1}]$ of the parallelizable spheres are algebraic quasigroups…
In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules $V$ of a semisimple quasi-Hopf algebra $H$ via the categorical counterpart developed in \cite{NS05}. We prove that this definition of higher FS-indicators coincides with the higher indicators introduced by Kashina, Sommerhäuser, and Zhu when $H$ is a Hopf algebra. We also obtain a sequence of canonical central elements of $H$, which is invariant under gauge transformations, whose values, when evaluated by the character of an $H$-module $V$, are the higher Frobenius-Schur indicators of $V$. As an …
We introduce quasi-Hopf $*$-algebras i.e. quasi-Hopf algebras equipped with a conjugation (star) operation. The definition of quasi-Hopf $*$-algebras proposed ensures that the class of quasi-Hopf $*$-algebras is closed under twisting and additionally, that any Hopf $*$-algebra becomes a quasi-Hopf $*$-algebra via twisting. The basic properties of these algebras are developed. The relationship between the antipode and star structure is investigated. Quasi-triangular quasi-Hopf $*$-algebras are introduced and studied.
We prove the quasi-Hopf algebra version of the Nichols-Zoeller theorem: A finite-dimensional quasi-Hopf algebra is free over any quasi-Hopf subalgebra.
The faithful quasi-dual $H^d$ and strict quasi-dual $H^{d'}$ of an infinite braided Hopf algebra $H$ are introduced and it is proved that every strict quasi-dual $H^{d'}$ is an $H$-Hopf module. The connection between the integrals and the maximal rational $H^{d}$-submodule $H^{d rat}$ of $H^{d}$ is found. That is, $H^{d rat}\cong \int ^l_{H^d} \otimes H$ is proved. The existence and uniqueness of integrals for braided Hopf algebras in the Yetter-Drinfeld category $(^B_B{\cal YD},C)$ are given.
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.
Let H be a finite-dimensional quasi-Hopf algebra. We show for each quotient quasibialgebra Q of H that Q is a quasi-Hopf algebra whose dimension divides the dimension of H.
We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra $D^ω(H)$ defined in \cite{bp} and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for $D^ω(H)$.
Let $m$ a positive integer, not divisible by 2,3,5,7. We generalize the classification of basic quasi-Hopf algebras over cyclic groups of prime order given in \cite{EG3} to the case of cyclic groups of order $m$. To this end, we introduce a family of non-semisimple radically graded quasi-Hopf algebras $A(H,s)$, constructed as subalgebras of Hopf algebras twisted by a quasi-Hopf twist, which are not twist equivalent to Hopf algebras. Any basic quasi-Hopf algebra over a cyclic group of order $m$ is either semisimple, or is twist equivalent to a Hopf algebra or a quasi-Hopf algebra of type $A(H,s…
We analyze some relations between quasi-Hopf smash products and certain twisted tensor products of quasialgebras. Along the way we obtain also some results of independent interest, such as a duality theorem for finite dimensional quasi-Hopf algebras and a universal property for generalized diagonal crossed products.
We define the notion of factorizable quasi-Hopf algebra by using a categorical point of view. We show that the Drinfeld double $D(H)$ of any finite dimensional quasi-Hopf algebra $H$ is factorizable, and we characterize $D(H)$ when $H$ itself is factorizable. Finally, we prove that any finite dimensional factorizable quasi-Hopf algebra is unimodular. In particular, we obtain that the Drinfeld double $D(H)$ is a unimodular quasi-Hopf algebra.
Let p and q be distinct odd primes and assume k is an algebraically closed field of characteristic zero. We classify all quasitriangular Hopf algebras of dimension pq^2 over k, which are not simple as Hopf algebras. Moreover, we obtained all quasitriangular structures on these Hopf algebras.
We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension $4p$ for a prime $p>5$. In particular, we obtain families of new examples of finite-dimensional Hopf algebras without the dual Chevalley property.
We put the known results on the antipode of a usual quasitriangular Hopf algebra into the framework of multiplier Hopf algebras. We illustrate with examples which can not be obtained using classical Hopf Algebras. The focus of the present paper lies on the class of the so-called group cograded multiplier Hopf algebras. By doing so, we bring the results of quasitriangular Hopf group-coalgebras (as introduced by Turaev) to the more general framework of multiplier Hopf algebras.
In [4], some quasi-Hopf algebras of dimension $n^{3}$, which can be understood as the quasi-Hopf analogues of Taft algebras, are constructed. Moreover, the quasi-Hopf analogues of generalized Taft algebras are considered in [7], where the language of the dual of a quasi-Hopf algebra is used. The Drinfeld doubles of such quasi-Hopf algebras are computed in this paper. The authors in [5] shew that the Drinfeld double of a quasi-Hopf algebra of dimension $n^{3}$ constructed in [4] is always twist equivalent to Lusztig's small quantum group $u_q(sl_2)$ if $n$ is odd. Based on computations and anal…
In this paper we show that weak Hopf (co)quasigroups can be characterized by a Galois-type condition. Taking into account that this notion generalizes the ones of Hopf (co)quasigroup and weak Hopf algebra, we obtain as a consequence the first fundamental theorem for Hopf (co)quasigroups and a characterization of weak Hopf algebras in terms of bijectivity of a Galois-type morphism (also called fusion morphism).
Let H be a quasi-Hopf algebra, a weak Hopf algebra or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v:H\rightarrow B. Then we can define an object B^{co(H)} which is a left-left Yetter-Drinfeld module over H, having extra properties that allow to make a smash product B^{co(H)}# H which is an H-bicomodule algebra, isomorphic to B.
Let $\mathds{k}$ be an algebraically closed field of characteristic zero. We determine all finite-dimensional Hopf algebras over $\mathds{k}$ whose Hopf coradical is isomorphic to the unique $12$-dimensional Hopf algebra $\mathcal{C}$ without the dual Chevalley property, such that the diagrams are strictly graded and the corresponding infinitesimal braidings are indecomposable objects in ${}_{\mathcal{C}}^{\mathcal{C}}\mathcal{YD}$. In particular, we obtain new Nichols algebras of dimension $18$ and $36$ and two families of new Hopf algebras of dimension $216$.
We recall the notion of a Hopf (co)quasigroup defined in \cite{Kl09} and define integration and Fourier Transforms on these objects analogous to those in the theory of Hopf algebras. Using the general Hopf module theory for Hopf (co)quasigroups from \cite{Br09} we show that a finite dimensional Hopf (co)quasigroup has a unique integration up to scale and an invertible antipode. We also supply the inverse Fourier transformation and show that it maps the convolution product on $H$ to the product in its dual $H^*$. Finally, we further develop the theory to consider Frobenius Hopf (co)quasigroups,…
In this paper, we present a general method for constructing finite-dimensional quasi-Hopf algebras from finite abelian groups and braided vector spaces of Cartan type. The study of such quasi-Hopf algebras leads to the classification of finite-dimensional radically graded basic quasi-Hopf algebras over abelian groups with dimensions not divisible by $2,3,5,7$ and associators given by abelian $3$-cocycles. As special cases , the small quasi-quantum groups are introduced and studied. Many new explicit examples of finite-dimensional genuine quasi-Hopf algebras are obtained.
We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension.
We present new Hopf algebras with the dual Chevalley property by determining all semisimple Hopf algebras Morita-equivalent to a group algebra over a finite group, for a list of groups supporting a non-trivial finite-dimensional Nichols algebra.
We show that two competing definitions of a ribbon quasi-Hopf algebra are actually equivalent. Along the way, we look at the Drinfel'd element from a new perspective and use this viewpoint to derive its fundamental properties.
A quasi-Hopf algebra $H$ can be seen as a commutative algebra $A$ in the centre $\mathcal Z(H-Mod)$ of $H-Mod$. We show that the category of $A$-modules in $\mathcal Z(H-Mod)$ is equivalent (as a monoidal category) to $H-Mod$. This can be regarded as a generalization of the structure theorem of Hopf bimodules of a Hopf algebra to the quasi-Hopf setting.
We introduce the notions of sovereign, spherical and balanced quasi-Hopf algebra. We investigate the connections between these, as well as their connections with the class of pivotal, involutory and ribbon quasi-Hopf algebras, respectively. Examples of balanced and ribbon quasi-Hopf algebras are obtained from a sort of double construction which associates to a braided category (resp. rigid braided) a balanced (resp. ribbon) one.
We show that bicrossed product Hopf algebras arising from exact factorizations in almost simple finite groups, so in particular, in simple and symmetric groups, admit no quasitriangular structure.
The class of graded elementary quasi-Hopf algebras of tame type is classified. Combining with our previous work [19], this completes the trichotomy for such class of algebras according to their representation types. In addition, new examples of genuine elementary quasi-Hopf algebras, and accordingly finite pointed tensor categories, are provided.
Let H be a finite-dimensional quasibialgebra. We show that H is a quasi-Hopf algebra if and only if the category of its finite-dimensional left modules is rigid if and only if a structure theorem for Hopf modules over H holds. We also show that a dual structure theorem for Hopf modules over a coquasibialgebra H holds if and only if the category of finite-dimensional right H-comodules is rigid; this is not equivalent to H being a coquasi-Hopf algebra.
In this paper we introduce the notion of weak Hopf quasigroup as a generalization of weak Hopf algebras and Hopf quasigroups. We obtain its main properties and we prove the fundamental theorem of Hopf modules for these algebraic structures.
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.
We construct quasi-Hopf algebras associated with a semisimple Lie algebra, a complex curve and a rational differential. This generalizes our previous joint work with V. Rubtsov (Israel J. Math. (1999) and q-alg/9608005).
We show that various cyclic and cocyclic modules attached to Hopf algebras and Hopf modules are related to each other via Connes' duality isomorphism for the cyclic category.
A detailed presentation of the results obtained during my Ph.D. research. The main investigations concern explicit descriptions of classes of finite dimensional pointed Hopf algebras and their quasi-isomorphism types.
Let $H$ be a quasitriangular quasi-Hopf algebra, we construct a braided group $\underline{H}$ in the quasiassociative category of left $H$-modules. Conversely, given any braided group $B$ in this category, we construct a quasi-Hopf algebra $B\rtimes H$ in the category of vector spaces. We generalise the transmutation and bosonisation theory of [10] to the quasi case. As examples, we bosonise the octonion algebra to an asoociative one, obtain the twisted quantum double $D^ϕ(G)$ of a finite group as a bosonisation, and obtain its transmutation. Finally, we show that $\underline{H}\rtimes H$ is i…
