We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an application, we give an alternative approach to Virelizier's version of the Fundamental Theorem for Hopf algebras. We introduce Yetter-Drinfeld modules over Hopf group-coalgebras using the center construction.
We study algebraic properties of Hopf group-coalgebras, recently introduced by Turaev. We show the existence of integrals and traces for such coalgebras, and we generalize the main properties of quasitriangular and ribbon Hopf algebras to the setting of Hopf group-coalgebras.
We provide an analog of the Drinfeld quantum double construction in the context of crossed Hopf group coalgebras introduced by Turaev. We prove that, provided the base group is finite, the double of a semisimple crossed Hopf group coalgebra is both modular and unimodular.
The center Z(C) of a spherical fusion category C (over an arbitrary commutative ring) is modular. We give an algorithm for computing the Reshetikhin-Turaev invariant defined with Z(C). It is based on Hopf diagrams and an explicit description of the structure of the coend of Z(C).
We provide an analog of Tannaka Theory for Hopf algebras in the context of crossed Hopf group coalgebras introduced by Turaev. Following Street and our previous work on the quantum double of crossed structures, we give a construction, via Tannaka Theory, of the quantum double of crossed Hopf group algebras (not necessarily of finite type).
In this paper we construct the Differential calculus on the Hopf Group Coalgebra introduced by Turaev [10]. We proved that the concepts introduced by S.L.Woronowicz in constructing Differential calculus on Hopf Compact Matrix Pseudogroups (Quantum Groups)[7] can be adapted to serve again in our construction.
In a recent paper the authors Beliakova, Blanchet and Gainutdinov have shown that the modified trace on the category $H$-pmod of the projective modules corresponds to the symmetrised integral on the finite dimensional pivotal Hopf algebra $H$. We generalize this fact to the context of $G$-graded categories and Hopf $G$-coalgebra studied by Turaev-Virelizier. We show that the symmetrised $G$-integral on a finite type pivotal Hopf $G$-coalgebra induces a modified trace in the associated $G$-graded category.
We provide an analog of the Joyal-Street center construction and of the Kassel-Turaev categorical quantum double in the context of the crossed categories introduced by Turaev. Then, we focus or attention to the case of categories of representation. In particular, we introduce the notion of a Yetter-Drinfeld module over a crossed group coalgebra H and we prove that both the category of Yetter-Drinfeld modules over H and the center of the category of representations of H as well as the category of representations of the quantum double of H are isomorphic as braided crossed categories.
We introduce Turaev bicategories and Turaev pseudofunctors. On the one hand, they generalize the notions of Turaev categories (and Turaev functors), introduced at the turn of the millennium and originally called "crossed group categories" by Turaev himself, and the notions of bicategories and pseudofunctors, on the other. For bimonads in 2-categories, which we defined in one of our previous papers, we introduce generalized Yetter-Drinfel`d modules in 2-categories. These generalize to the 2-categorical setting the generalized Yetter-Drinfel`d modules (over a field) of Panaite and Staic, and thu…
We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer's Lie coalgebra of trees, and extend this to a map from a certain quiver-theoretic Hopf algebra to Connes and Kreimer's renormalization Hopf algebra, as well as to pre-Lie versions. These results are direct analogues of Turaev's results in 2004, by replacing algebras of loops on surfaces with algebras of paths on quivers. We also factor the morphism through an algebra of chord diagrams and explain the geometric version. We then explain how all of the Hopf algebras are uniquely determin…
We give a categorical setting in which Penrose graphical calculus naturally extends to graphs drawn on the boundary of a handlebody. We use it to introduce invariants of 3-manifolds presented by Heegaard splittings. We recover Kuperberg invariants when the category comes from an involutory Hopf algebra and Turaev-Viro invariants when the category is semi-simple and spherical.
The Multiplier Hopf Group Coalgebra was introduced by Hegazi in 2002 [7] as a generalization of Hope group caolgebra, introduced by Turaev in 2000 [5], in the non-unital case. We prove that the concepts introduced by A.Van Daele in constructing multiplier Hopf algebra \cite{4} can be adapted to serve again in our construction. A multiplier Hopf group coalgebra is a family of algebras $A=\{A_α\}_{α\in π}$, ($π$ is a discrete group) equipped with a family of homomorphisms $Δ=\{Δ_{α,β}:A_{αβ}\longrightarrow M(A_α\otimes A_β)\}_{α,β\in π}$ which is called a comultiplication under some conditions, …
Let $G$ be a group and assume that $(A_p)_{p\in G}$ is a family of algebras with identity. We have a {\it Hopf $G$-coalgebra} (in the sense of Turaev) if, for each pair $p,q\in G$, there is given a unital homomorphism $\co_{p,q}:A_{pq}\to A_p \ot A_q$ satisfying certain properties. Consider now the direct sum $A$ of these algebras. It is an algebra, without identity, except when $G$ is a finite group, but the product is non-degenerate. The maps $\co_{p,q}$ can be used to define a coproduct $\co$ on $A$ and the conditions imposed on these maps give that $(A,\co)$ is a multiplier Hopf algebra. I…
The quantum co-ordinate algebra $A_{q}(\mathfrak{g})$ associated to a Kac-Moody Lie algebra $\mathfrak{g}$ forms a Hopf algebra whose comodules are precisely the $U_{q}(\mathfrak{g})$ modules in the BGG category $\mathcal{O}_{\mathfrak{g}}$. In this paper we investigate whether an analogous result is true when $q=0$. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this we prove that there is no coalgebra in the category of pointed sets whose comodules are equiva…
We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The self-distributive operations of these structures provide solutions of the Yang--Baxter equation, and, conversely, solutions of the Yang--Baxter equation can be used to construct self-distributive operations in certain categories. Moreover, we present a cohomology theory that encompasses both Lie algebra and quandle cohomologies, is …
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.
Let H be a finite-dimensional pivotal and unimodular Hopf algebra over a field k. It was shown in [BBGa] that the projective tensor ideal in H-mod admits a unique non-degenerate modified trace, a natural generalisation of the categorical trace. This paper provides an extension of this result to a much more general setting. We first extend the notion of the modified trace to the so-called module trace for a given k-linear module category M over a pivotal category C equipped with a module endofunctor. We provide a non-trivial class of examples of such module traces. In particular, we show that a…
Suppose that we have a semisimple, connected, simply connected algebraic group $G$ with corresponding Lie algebra $\mathfrak{g}$. There is a Hopf pairing between the universal enveloping algebra $U(\mathfrak{g})$ and the coordinate ring $O(G)$. By introducing a parameter $q$, we can consider quantum deformations $U_q(\mathfrak{g})$ and $O_q(G)$ respectively, between which there again exists a Hopf pairing. We show that the category of crystals associated with $U_q(\mathfrak{g})$ is a monoidal category. We define subgroups of $U_q(\mathfrak{g})$ to be right coideal subalgebras, and subgroups of…
Motivated by ideas from string theory and quantum field theory new invariants of knots and 3-dimensional manifolds have been constructed from complex algebraic structures such as Hopf algebras (Reshetikhin and Turaev), monoidal categories with additional structure (Turaev and Yetter), and modular functors (Walker and Kontsevich). These constructions are very closely related. We take a unifying categorical approach based on a natural 2-dimensional generalization of a topological field theory in the sense of Atiyah and Segal, and show that the axioms defining these complex algebraic structures a…
Let $G$ be any group and let $K(G)$ denote the multiplier Hopf algebra of complex functions with finite support in $G$. The product in $K(G)$ is pointwise. The comultiplication on $K(G)$ is defined with values in the multiplier algebra $M(K(G) \otimes K(G))$ by the formula $(Δ(f)) (p,q) = f(pq)$ for all $f \in K(G)$ and $p, q \in G$. In this paper we consider multiplier Hopf algebras $B$ (over $\Bbb C$) such that there is an embedding $I: K(G) \to M(B)$. This embedding is a non-degenerate algebra homomorphism which respects the comultiplication and maps $K(G)$ into the center of $M(B)$. These …
In $1998$, Connes and Moscovici defined the cyclic cohomology of Hopf algebras. In $2010$, Khalkhali and Pourkia proposed a braided generalization: to any Hopf algebra $H$ in a braided category $\mathcal B$, they associate a paracocyclic object in $\mathcal B$. In this paper we explicitly compute the powers of the paracocyclic operator of this paracocyclic object. Also, we introduce twisted modular pairs in involution for $H$ and derive (co)cyclic modules from them. Finally, we relate the paracocyclic object associated with $H$ to that associated with an $H$-module coalgebra via a categorical …
For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion is formulaic while the latter is categorical. We generalize this result to the setting of weak bialgebras $H$. The category $\mathcal{M}^H$ admits a monoidal structure by work of Nill and Böhm-Caenepeel-Janssen, but the algebras in $\mathcal{M}^H$ are not canonically $\Bbbk$-algebras. Nevertheless, we prove that there is an isomorphism between the category o…
We develop an algebraic theory of colored, semigrouplike-flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in combinatorics, topology, number theory and physics. In particular, we can precisely give conditions for the invertibility of characters that is needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. In order to construct antipodes, we discuss formal localization constructions and quantum deformations. These allow to define and explain the ap…
In this paper, we prove that there is a canonical homotopy $(n+1)$-algebra structure on the shifted operadic deformation complex $Def(e_n\to\mathcal{P})[-n]$ for any operad $\mathcal{P}$ and a map of operads $f\colon e_n\to\mathcal{P}$. This result generalizes the result of [T2], where the case $\mathcal{P}=\mathrm{End}_{Op}(X)$ was considered. Another more computational proof of the same statement was recently sketched in [CW]. Our method combines the one of [T2] with the categorical algebra on the category of symmetric sequences, introduced in [R] and further developed in [KM] and [Fr1]. We …
Hopf algebroids are generalization of Hopf algebras over non-commutative base rings. It consists of a left- and a right-bialgebroid structure related by a map called the antipode. However, if the base ring of a Hopf algebroid is commutative one does not necessarily have a Hopf algebra. Meanwhile, a Hopf category is the categorification of a Hopf algebra. It consists of a category enriched over a braided monoidal category such that every hom-set carries a coalgebra structure together with an antipode functor. In this article, we will introduce the notion of a topological Hopf category$-$ a smal…
We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{iπ/\ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C^*-category of the category of tilting modules of the non-semisimple quantum group U_q({\mathfrak sl}_N) of Drinfeld, Jimbo and Lusztig. As an algebra, the C^*-groupoid is a quotient of U_q({\mathfrak sl}_N). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociati…
The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. In order to do that we define the Hochschild cohomology of an algebra in an abelian monoidal category. Then we characterize those algebras which have dimension less than or equal to 1 with respect to Hochschild cohomology. Now let us consider a Hopf algebra A such that its Jacobson radical J is a nilpotent Hopf ideal and H:=A/J is a semisimple algebra. By using our homological results, we prove that the canonical proje…
