The notion of vertex operator coalgebra is presented which corresponds to the family of correlation functions of one string propagating in space-time splitting into n strings in conformal field theory. This notion is in some sense dual to the notion of vertex operator algebra. We prove that any vertex operator algebra equipped with a non-degenerate, Virasoro preserving, bilinear form gives rise to a corresponding vertex operator coalgebra.
We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on vertex coalgebras. The former two properties require grading. We then discuss comodule structure. We conclude by discussing instances where graded vertex coalgebras appear, particularly as related to Primc's vertex Lie algebra and (universal) enveloping vertex algebras.
Based on the definition of vertex coalgebra introduced by Hubbard [H], we prove that this notion can be reformulated using the Co-Commutator, Co-Skew symmetry and Co-Associator formulas without restrictions on the grading.
The notion of vertex operator coalgebra is presented and motivated via the geometry of conformal field theory. Specifically, we describe the category of geometric vertex operator coalgebras, whose objects have comultiplicative structures meromorphically induced by conformal equivalence classes of worldsheets. We then show this category is isomorphic to the category of vertex operator coalgebras, which is defined in the language of formal algebra. The latter has several characteristics which give it the flavor of a coalgebra with respect to the structure of a vertex operator algebra and several…
In this paper we study oriented quantum coalgebras which are structures closely related to oriented quantum algebras. We study the relationship between oriented quantum coalgebras and oriented quantum algebras and the relationship between oriented quantum coalgebras and quantum coalgebras. We show that there are regular isotopy invariants of oriented 1-1 tangles and of oriented knots and links associated to oriented and twist oriented quantum coalgebras respectively. There are many parallels between the theory of oriented quantum coalgebras and the theory of quantum coalgebras
This notes gives a proof that a connected coaugmented cofiltered coalgebra is a conilpotent coalgebra and thus a connected coaugmented cofiltered bialgebra is a Hopf algebra. This applies in particular to a connected coaugmented cograded coalgebra and a connected coaugmented cograded bialgebra.
In this paper, we give several necessary conditions for non-cosemisimple coalgebras being admissible. The implications simplify the classification problems for Hopf algebras of dimension 45, 105 and a few others.
We give the explicit construction of the product of an arbitrary family of coalgebras, bialgebras and Hopf algebras: it turns out that the product of an arbitrary family of coalgebras (resp. bialgebras, Hopf algebras) is the sum of a family of coalgebras (resp. bialgebras, Hopf algebras). The equalizers of two morphisms of coalgebras (resp. bialgebras, Hopf algebras) are also described explicitly. As a consequence the categories of coalgebras, bialgebras and Hopf algebras are shown to be complete and a explicit description for limits in the above categories is given.
The aim of this paper is to generalize the concept of Lie-admissible coalgebra introduced by Goze and Remm to Hom-coalgebras and to introduce Hom-Hopf algebras with some properties. These structures are based on the Hom-algebra structures introduced by the authors.
In this work the notions of partial action of a weak Hopf algebra on a coalgebra and partial action of a groupoid on a coalgebra will be introduced, just as some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of partial action on a coalgebra and partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.
We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. A dual version of Brauer's equivalence theorem is presented, allowing a characterization of symmetric coalgebras by comparing certain functors. We define an automorphism of the ring with local units constructed from a co-Frobenius coalgebra, which we call the Nakayama automorphism. This is used to give a new characterization to symmetric c…
Universal measuring coalgebras provide an enrichment of the category of algebras over the category of coalgebras. By considering the special case of the tensor algebra on a vector space V, the category of linear spaces itself becomes enriched over coalgebras, and the universal measuring coalgebra is the dual coalgebra of the tensor algebra T(V tensor V*). Given a braiding R on V the universal measuring coalgebra P_R(V) which preserves the grading is naturally dual to the Fadeev-Takhtadjhan-Reshitikin bialgebra A(R) and therefore provides a representation of the quantized universal enveloping a…
We investigate the notion of associated graded coalgebra (algebra) of a bialgebra with respect to a subbialgebra (quotient bialgebra) and characterize those which are bialgebras of type one in the framework of abelian braided monoidal categories.
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.
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, …
The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom-Lie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom-Hopf algebras and prove some fundamental properties and give examples. Finally, we define the concept of Hom-Lie admissible Hom-coalgebra an…
In order to construct solutions of the braid equation we consider bijective left non-degenerate set-theoretic type solutions, which correspond to regular q-cycle coalgebras. We obtain a partial classification of the different q-cycle coalgebra structures on the dual coalgebra of $K[y]/\langle y^n\rangle$, the truncated polynomial algebra. We obtain an interesting family of involutive q-cycle coalgebras which we call Standard Cycle Coalgebras. They are parameterized by free parameters $\{p_1,...,p_{n-1}\}$ and in order to verify that they are compatible with the braid equation, we have to verif…
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).
We construct from coassociative coalgebras, bialgebras, Hopf algebras, new objects such as Poisson algebras, Leibniz algebras defined by J-L Loday and M. Ronco and explore the notion of coassociative manifolds.
We construct the analogue of Takeuchi's free Hopf algebra in the setting of Poisson Hopf algebras. More precisely, we prove that there exists a free Poisson Hopf algebra on any coalgebra or, equivalently that the forgetful functor from the category of Poisson Hopf algebras to the category of coalgebras has a left adjoint. In particular, we also prove that the category of Poisson Hopf algebras is a reflective subcategory of the category of Poisson bialgebras. Along the way, we describe coproducts and coequalizers in the category of Poisson Hopf algebras, therefore showing that the latter catego…
For a symmetry braid S-Lie coalgebras, as a dual object to algebras introduced by Gurevich, are considered. For an Young antisymmetrizer an S-exterior algebra is introduced. From this differential point of view S-Lie coalgebras are investigated. The dual Koszul theorem in this case is proved.
The notion of a coalgebra-Galois extension is defined as a natural generalisation of a Hopf-Galois extension. It is shown that any coalgebra-Galois extension induces a unique entwining map $ψ$ compatible with the right coaction. For the dual notion of an algebra-Galois coextension it is also proven that there always exists a unique entwining structure compatible with the right action.
In this note we study dual coalgebras of algebras over arbitrary (noetherian) commutative rings. We present and study a generalized notion of coreflexive comodules and use the results obtained for them to characterize the so called coreflexive coalgebras. Our approach in this note is an algebraically topological one.
We generalize the results on existence and uniqueness of integrals from compact groups and Hopf algebras in a pure (co)algebraic setting, and find a series of new results on (quasi)-co-Frobenius and semiperfect coalgebras. For a coalgebra $C$, we introduce the generalized space of integrals $\int_M=\Hom^C(C,M)$ associated to a right $C$-comodule $M$ and study connections between "uniqueness of integrals" $\dim(\int_M)\leq \dim(M)$ and "existence of integrals" $\dim(\int_M)\geq \dim(M)$ for all $M$ and representation theoretic properties of $C$: (quasi)-co-Frobenius, semiperfect. We show that a…
In this note we extend the cyclic homology functor, and in particular the periodic cyclic homology, to the category of DG (= differential graded) coalgebras. We are partly motivated by the question of products and coproducts in the cyclic homology of algebras, pioneered by A.Connes. As an application we show how one can start from the classical shuffle map in homological algebra and algebraic topology, interpreted by Husemoller, Moore and Stasheff as a morphism of DG coalgebras, and build a theory of products and coproducts in periodic and negative cyclic homology. One of the key ingredients i…
We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed oriented manifold, and a gravity coalgebra structure on the reduced equivariant homology.
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 introduce the notion of a crossed product of an algebra by a coalgebra $C$, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right $C$-comodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimensional quantum Euclidean group is a coalgebra crossed product. The paper is completed with an appendix describing the dualisation of construction of coalgebra crossed products.
We consider the factorisation problem for bialgebras: when a bialgebra $K$ factorises as $K=HL$, where $H$ and $L$ are algebras and coalgebras (but not necessarly bialgebras). Given two maps $R: H\ot L\to L\ot H$ and $W:L\ot H\to H\ot L$, we introduce a product $L_W\bowtie_R H$ and give necessary and sufficient conditions for $L_W\bowtie_R H$ to be a bialgebra. It turns out that $K$ factorises as $K=HL$ if and only if $K\cong L_W\bowtie_R H$ for some maps $R$ and $W$. As examples of this product we recover constructions introduced by Majid and Radford. Also, some of the pointed Hopf algebras t…
We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices $M_2(\C)=\C\Z_2\cdot\C\Z_2$. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct $q$-monopoles on all the Podleś quantum spheres $S^2_{q,s}$.
This paper is concerned with the theory of cup-products in Hopf-type cyclic cohomology of algebras and coalgebras. Here we give detailed proofs of the statements, announced in our previous paper. We show that the cyclic cohomology of a coalgebra can be obtained from a construction involving noncommutative Weil algebra. Then we use a generalization of Quillen and Crainic's construction to define the cup-product. We discuss the relation of the introduced cup-product and $S$-operations on cyclic cohomology. After this we describe the relation of this type of product and bivariant cyclic cohomolog…
We study non-counital coalgebras and their dual non-unital algebras, and introduce the finite dual of a non-unital algebra. We show that a theory that parallels in good part the duality in the unital case can be constructed. Using this, we introduce a new notion of left coreflexivity for counital coalgebras, namely, a coalgebra is left coreflexive if $C$ is isomorphic canonically to the finite dual of its left rational dual $Rat(_{C^*}C^*)$. We show that right semiperfectness for coalgebras is in fact essentially equivalent to this left reflexivity condition, and we give the connection to usua…
We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same cod…
In this work we study the induction (induced and coinduced)theory for Hopf group coalgebra. We define a substructure B of a Hopf group coalgebra $H$, called subHopf group coalgebra. Also, we have introduced the definition of Hopf group suboalgebra and group coisotropic quantum subgroup of $H$. The properties of the algebraic structure of the induced and coinduced are given. Moreover, a framework of the geometric interperation and simplicity theory of such representation strructure are stuided.
We study new coalgebra structures on the tensor product of two coalgebras $C$ and $D$ by twisting the tensor product coalgebra via a twist map $Ψ: C \otimes D \rightarrow D \otimes C$. We deal with the general case in which the counit of the tensor product coalgebra is deformed as well. Some classes of such deformations are analyzed and a notion of equivalence of twists is discussed. We also present the dual deformation of tensor product algebras and provide examples.
An associative algebra is nothing but an odd quadratic codifferential on the tensor coalgebra of a vector space, and an A-infinity algebra is simply an arbitrary odd codifferential. Hochschild cohomology classifies the deformations of an associative algebra into an A-infinity algebra, and cyclic cohomology in the presence of an invariant inner product classifies the deformations of the associative algebra into an A-infinity algebra preserving the inner product. Similarly, a graded Lie algebra is simply a special case of an odd codifferential on the exterior coalgebra of a vector space, and an …
In this article -that has also the intention to survey some known results in the theory of compact quantum groups using methods different from the standard and with a strong algebraic flavor- we consider compact o-coalgebras and Hopf algebras. In the case of a o-Hopf algebra we present a proof of the characterization of the compactness in terms of the existence of a positive definite integral, and use our methods to give an elementary proof of the uniqueness - up to conjugation by an automorphism of Hopf algebras- of the compact involution appearing in [4]. We study the basic properties of the…
A duality transform for the coalgebra of the free difference quotient derivation-multiplication of an operator with respect to a free algebra of scalars is constructed. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalization of the difference quotient derivation.
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 $Γ$ …
Two cochain complexes are constructed for an algebra A and a coalgebra C entwined with each other via the map $ψ:C\otimes A\to A\otimes C$. One complex is associated to an A-bimodule, the other to a C-bicomodule. In the former case the resulting complex can be considered as a $ψ$-twisted Hochschild complex of A, while for the latter one obtains a $ψ$-twist of the Cartier complex of C. The notion of a {\em weak comp algebra} is introduced by weakening the axioms of the Gerstenhaber comp algebra. It is shown that such a weak comp algebra is a cochain complex with two cup products that descend to…
For a given Hopf algebra $A$ we classify all Hopf algebras $E$ that are coalgebra split extensions of $A$ by $H_4$, where $H_4$ is the Sweedler's 4-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras $A # H_4$ by computing explicitly two classifying objects: the cohomological 'group' ${\mathcal H}^{2} (H_4, A)$ and $\textsc{C}\textsc{r}\textsc{p} (H_4, A) :=$ the set of types of isomorphisms of all crossed products $A # H_4$. All crossed products $A #H_4$ are described by generators and relations and classified: they are parameterized by the set ${\mathcal…
We give an algebraic construction of the topological graph-tree configuration pairing of Sinha and Walter beginning with the classical presentation of Lie coalgebras via coefficients of words in the associative Lie polynomial. Our work moves from associative algebras to preLie algebras to graph complexes, justifying the use of graph generators for Lie coalgebras by iteratively expanding the set of generators until the set of relations collapses to two simple local expressions. Our focus is on new computational methods allowed by this framework and the efficiency of the graph presentation in pr…
The incidence algebra of a partially ordered set (poset) supports in a natural way also a coalgebra structure, so that it becomes a m-weak bialgebra even a m-weak Hopf algebra with Möbius function as antipode. Here m-weak means that multiplication and comultiplication are not required to be coalgebra- or algebra-morphisms, respectively. A rich theory is obtained in computing modulo an equivalence relation on the set of intervals in the poset.
We investigate left and right co-Frobenius coalgebras and give equivalent characterizations which prove statements dual to the characterizations of Frobenius algebras. We prove that a coalgebra is left and right co-Frobenius if and only if C is isomorphic to Rat(C*) as right C*-modules and also that this is eqhivalent to the fact that the functors Hom_K(-,K) and Hom_{C*}(-,C*) from the category of right C-comodules to the category of all left C*-modules are isomorphic. This allows a definition of a left-right symmetric concept of co-Frobenius coalgebras that is perfectly dual to the one of Fro…
We give a systematic construction of Hopf algebra structures on braided cofree coalgebras. The relevant underlying structures are braided algebras and braided coalgebras. We provide some interesting examples of these algebras and coalgebras related to quantum groups. We introduce quantum multi-brace algebras which are generalizations of both braided algebras and $\textbf{B}_\infty$-algebras, as the natural framework. This new subject enables one to quantize some important algebra structures in a uniform way. Particular interesting examples are quantum quasi-shuffle algebras.
We introduce two constructions of a coassociative comultiplication in the algebra of phrases in a given alphabet. As a preliminary step we give two constructions of a pre-Lie comultiplication in the module generated by words.
We introduce the notion of a partial corepresentation of a given Hopf algebra $H$ over a coalgebra $C$ and the closely related concept of a partial $H$-comodule. We prove that there exists a universal coalgebra $H^{par}$, associated to the original Hopf algebra $H$, such that the category of regular partial $H$-comodules is isomorphic to the category of $H^{par}$-comodules. We introduce the notion of a Hopf coalgebroid and show that the universal coalgebra $H^{par}$ has the structure of a Hopf coalgebroid over a suitable coalgebra.
"Co-Frobenius" coalgebras were introduced as dualizations of Frobenius algebras. Recently, it was shown in \cite{I} that they admit left-right symmetric characterizations analogue to those of Frobenius algebras: a coalgebra $C$ is co-Frobenius if and only if it is isomorphic to its rational dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; in the first main result we show that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently right) rational dual $Rat(C^*)$, in the sense that certain coproduct or prod…
OCHA is the homotopy algebra of open-closed strings. It can be defined as a sequence of multilinear operations on a pair of DG spaces satisfying certain relations which include the $L_\infty$ relations in one space and the $A_\infty$ relations in the other. In this paper we show that the OCHA structure is intrinsic to the tensor product of the symmetric and tensor coalgebras. We also show how an OCHA can be obtained from $A_\infty$-extesions and define the {\it universal enveloping} $A_\infty$-algebra of an OCHA as an $A_\infty$-extension of the universal enveloping of its $L_\infty$ part by i…
We develop the theory of special biserial and string coalgebras and other concepts from the representation theory of quivers. These tools are then used to describe the finite dimensional comodules and Auslander-Reiten quiver for the coordinate Hopf algebra of quantum SL(2) at a root of unity. We also describe the stable Green ring and compute quantum dimensions.
Completely integrable Hamiltonians defining classical mechanical systems of $N$ coupled oscillators are obtained from Poisson realizations of Heisenberg--Weyl, harmonic oscillator and $sl(2,\R)$ coalgebras. Various completely integrable deformations of such systems are constructed by considering quantum deformations of these algebras. Explicit expressions for all the deformed Hamiltonians and constants of motion are given, and the long-range nature of the interactions is shown to be linked to the underlying coalgebra structure. The relationship between oscillator systems induced from the $sl(2…
