In our previous work (math/0008128), we studied the set Quant(K) of all universal quantization functors of Lie bialgebras over a field K of characteristic zero, compatible with duals and doubles. We showed that Quant(K) is canonically isomorphic to a product G_0(K) \times Sha(K), where G_0(K) is a universal group and Sha(K) is a quotient set of a set B(K) of families of Lie polynomials by the action of a group G(K). We prove here that G_0(K) is equal to the multiplicative group 1 + h K[[h]]. So Quant(K) is `as close as it can be' to Sha(K). We also show that the only universal derivations of Lie bialgebras are multiples of the composition of the bracket with the cobracket. Finally, we prove that the stabilizer of any element of B(K) is reduced to the 1-parameter subgroup of G(K) generated by the corresponding `square of the antipode'.
Weak (Hopf) bialgebras are described as (Hopf) bimonoids in appropriate duoidal (also known as 2-monoidal) categories. This interpretation is used to define a category wba of weak bialgebras over a given field. As an application, the "free vector space" functor from the category of small categories with finitely many objects to wba is shown to possess a right adjoint, given by taking (certain) group-like elements. This adjunction is proven to restrict to the full subcategories of groupoids and of weak Hopf algebras, respectively. As a corollary, we obtain equivalences between the category of s…
The aim of this paper is to define and study Yetter-Drinfeld modules over Hom-bialgebras, a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism. Yetter-Drinfeld modules over a Hom-bialgebra with bijective structure map provide solutions of the Hom-Yang-Baxter equation. The category $_H^H{\mathcal YD}$ of Yetter-Drinfeld modules with bijective structure maps over a Hom-bialgebra H with bijective structure map can be organized, in two different ways, as a quasi-braided pre-tensor category. If H is quasitriangular (respectively coquasitri…
We continue the study of the representation theory of a regular weak multiplier bialgebra with full comultiplication, started in arXiv:1306.1466, arXiv:1311.2730. Yetter-Drinfeld modules are defined as modules and comodules, with compatibility conditions that are equivalent to a canonical object being (weakly) central in the category of modules, and equivalent also to another canonical object being (weakly) central in the category of comodules. Yetter-Drinfeld modules are shown to constitute a monoidal category via the (co)module tensor product over the base (co)algebra. Finite dimensional Yet…
By a theorem of Majid, every monoidal category with a neutral quasi-monoidal functor to finitely-generated and projective $\Bbbk$-modules gives rise to a coquasi-bialgebra. We prove that if the category is also rigid, then the associated coquasi-bialgebra admits a preantipode, providing in this way an analogue for coquasi-bialgebras of Ulbrich's reconstruction theorem for Hopf algebras. When $\Bbbk$ is field, this allows us to characterize coquasi-Hopf algebras as well in terms of rigidity of finite-dimensional corepresentations.
This is a sequel paper of arXiv:1306.1466 in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to …
This paper is a continuation of "Quantization of Lie bialgebras, I" (q-alg/9606005). We show that the quantization procedure defined in "Quantization of Lie bialgebras, I" is given by universal acyclic formulas and defines a functor from the category of Lie bialgebras to the category of quantized universal enveloping algebras. We also show that this functor defines an equivalence between the category of Lie bialgebras over k[[h]] and the category quantized universal enveloping (QUE) algebras.
A PROP is a symmetric monoidal category, whose set of objects is the set of natural numbers and on objects the monoidal structure is given by the addition. An algebra over a PROP is a symmetric strict monoidal functor to the tensor category of vector spaces. We give an explicite construction of the PROP whose category of algebras is equivalent to the category of bialgebras (= associative and coassociative bialgebras).
We study monoidal structures on the category of (co)modules over a weak bialgebra. Results due to Nill and Szlachányi are unified and extended to infinite algebras. We discuss the coalgebra structure on the source and target space of a weak bialgebra.
We prove that a finite-dimensional Hopf algebra with the dual Chevalley Property over a field of characteristic zero is quasi-isomorphic to a Radford-Majid bosonization whenever the third Hochschild cohomology group in the category of Yetter-Drinfeld modules of its diagram with coefficients in the base field vanishes. Moreover we show that this vanishing occurs in meaningful examples where the diagram is a Nichols algebra.
Central bialgebras in a braided category $\C$ are algebras in the center of the category of coalgebras in $\C$. On these bialgebras another product can be defined, which plays the role of the opposite product. Hence, coquasitriangular structures on central bialgebras can be defined. We prove some properties of the antipode on coquasitriangular central Hopf algebras and give a characterization of central bialgebras.
This paper is a continuation of the series of papers "Quantization of Lie bialgebras (QLB) I-V". We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with th…
We settle several questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. An important new technical ingredient introduced in this paper is an endofunctor D in the category of augmented props with the property that for any representation of a prop P in a vector space V the associated prop DP admits an induced representation on the graded commutative tensor algebra S(V) given in terms of polydifferential operators. Applying this functor to the prop LieB of Lie bialgebras we show that universal formali…
In this paper, the structure of cocommutative vertex bialgebras is investigated. For a general vertex bialgebra $V$, it is proved that the set $G(V)$ of group-like elements is naturally an abelian semigroup, whereas the set $P(V)$ of primitive elements is a vertex Lie algebra. For $g\in G(V)$, denote by $V_g$ the connected component containing $g$. Among the main results, it is proved that if $V$ is a cocommutative vertex bialgebra, then $V=\oplus_{g\in G(V)}V_g$, where $V_{\bf 1}$ is a vertex subbialgebra which is isomorphic to the vertex bialgebra ${\mathcal{V}}_{P(V)}$ associated to the ver…
Assume $k$ is a field and let $F:C\rightarrow Vect_{k}$ be a small $k$-linear functor from a $k$-linear abelian category $C$ to the category of vector spaces over the field $k$, the purpose of this note is to use a little knowledge of linear algebra and category to give the description of $end(F)$ and $coend(F)$, and then we give the reconstruction theorem of bialgebras by using this description. We use a constructive approach to understand $end(F),coend(F)$ and we describe the bialgebra structure of $coend(F)$ concretely when $F$ is a tensor functor.
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 solve a functional version of the problem of twist quantization of a coboundary Lie bialgebra (g,r,Z). We derive from this the following results: (a) the formal Poisson manifolds g^* and G^* are isomorphic; (b) we construct a subalgebra of U(g^*), isomorphic to S(g^*)^g. When g can be quantized, we construct a deformation of the morphism S(g^*)^g subset U(g^*). When g is quasitriangular and nondegenerate, we compare our construction with Semenov-Tian-Shansky's construction of a commutative subalgebra of U(g^*). We also show that the canonical derivation of the function ring of G^* is Hamilt…
A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the `base algebras') are shown to carry coseparable co-Frobenius coalgebra structures. Appropriate modules over a weak multiplier bialgebra are shown to constitute a monoidal category via the (co)module tensor product over the base algebra. The relation to Van Daele and Wang's (regular and arbitrary) weak multiplier Hopf algebra is discussed.
We study Thomas Willwacher's twisting endofunctor tw in the category of dg properads P under the operad of (strongly homotopy) Lie algebras. It is proven that if P is a properad under properad Lieb of Lie bialgebras , then the associated twisted properad tw(P) becomes in general a properad under quasi-Lie bialgebras (rather than under Lieb). This result implies that the cyclic cohomology of any cyclic homotopy associative algebra has in general an induced structure of a quasi-Lie bialgebra. We show that the cohomology of the twisted properad tw(Lieb) is highly non-trivial -- it contains the co…
Let H be a bialgebra and D an H-bimodule algebra H-bicomodule coalgebra. We find sufficient conditions on D for the L-R-smash product algebra and coalgebra structures on D\otimes H to form a bialgebra (in this case we say that (H, D) is an L-R-admissible pair), called L-R-smash biproduct. The Radford biproduct is a particular case, and so is, up to isomorphism, a double biproduct with trivial pairing. We construct a prebraided monoidal category {\cal LR}(H), whose objects are H-bimodules H-bicomodules M endowed with left-left and right-right Yetter-Drinfeld module as well as left-right and rig…
We construct a bialgebra object in the category of linear maps LM from a cocommutative rack bialgebra. The construction does extend to some non-cocommutative rack bialgebras, as is illustrated by a concrete example. As a separate result, we show that the Loday complex with adjoint coefficients embeds into the rack bialgebra deformation complex for the rack bialgebra defined by a Leibniz algebra.
For a C$^{*}$-bialgebra $A$ with a comultiplication $Δ$, a universal $R$-matrix of $(A,Δ)$ is defined as a unitary element in the multiplier algebra $M(A\otimes A)$ of $A\otimes A$ which is an intertwiner between $Δ$ and its opposite comultiplication $Δ^{op}$. We show that there exists no universal $R$-matrix for some C$^{*}$-bialgebras.
This preprint contains a part of the results of our earlier preprint arXiv:0907.3335v2 presented in a form suitable for journal publication. It covers a construction of a 2-fold monoidal structure on the category of tetramodules, with all necessary definitions, and an overview of the results of R.Taillefer [Tai1,2] on tetramodules and the Gerstenhaber-Schack cohomology [GS] (formerly served as Appendix in arXiv:0907.3335v2), as well as a computation of the Gerstenhaber-Schack cohomology for the free commutative cocommutative bialgebra S(V), for a V is a vector space.
The paper deals with three topics on coquasitriangular bialgebras. A characterization of universal r-forms in terms of Yetter-Drinfeld modules is given. All universal r-forms for the coordinate Hopf algebras of the quantum groups GL_q(N), SL_q(N), O_q(N) and Sp_q(N) are described. It is well-known that the square of the antipode of a coquasitriangular Hopf algebra A with universal r-form r can be expressed by means of the linear functional f_r(a)=r(a_{(1)}\otimes S(a_{(2)})), a\in A. For the standard compact matrix groups the functional f_r is related to the Woronowicz modular character f_{-2}…
One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category Vect. In the past this problem has been resolved by working with a weaker structure called a `twisted' bialgebra. In this paper we solve the problem differently by first switching to a different underlying category Vect^K of vector spaces graded by a group K called the Grothendieck group. We equip this category with a nontrivial braiding which depends on the K-grading. With this braidin…
We construct quantization functors of quasi-Lie bialgebras. We establish a bijection between this set of quantization functors, modulo equivalence and twist equivalence, and the set of quantization functors of Lie bialgebras, modulo equivalence. This is based on the acyclicity of the kernel of the natural morphism from the universal deformation complex of quasi-Lie bialgebras to that of Lie bialgebras. The proof of this acyclicity consists in several steps, ending up in the acyclicity of a complex related to free Lie algebras, namely, the universal version of the Lie algebra cohomology complex…
The main purpose of this paper is to study Quantization of color Lie bialgebras, generalizing to color case the approach by Etingof-Kazhdan which were considered for superbialgebras by Geer. Moreover we discuss Drinfeld category, Quantization of Triangular color Lie bialgebras and Simple color Lie bialgebras of Cartan type.
Let A be a finite dimensional unital associative algebra over a field K, which is also equipped with a coassociative counital coalgebra structure (Δ,\eps). A is called a Weak Bialgebra if the coproduct Δis multiplicative. We do not require Δ(1) = 1 \otimes 1 nor multiplicativity of the counit \eps. Instead, we propose a new set of counit axioms, which are modelled so as to guarantee that \Rep\A becomes a monoidal category with unit object given by the cyclic A-submodule \E := (A --> \eps) \subset \hat A (\hat A denoting the dual weak bialgebra). Under these monoidality axioms \E and \bar\E := …
All coboundary Lie bialgebras and their corresponding Poisson--Lie structures are constructed for the oscillator algebra generated by $\{å,\ap,\am,\bb\}$. Quantum oscillator algebras are derived from these bialgebras by using the Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both algebra and group levels are obtained, including their universal $R$--matrices.
We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence. In the second step one deformation quantizes a quantizable Poisson (resp. Lie bialgebra) structure. We show again explicit transcendental formulae for this second step correspondence (as a byproduct we obtain configuration space models for biassociahedron and bipermut…
We introduce a new type of categorical object called a \emph{hom-tensor category} and show that it provides the appropriate setting for modules over an arbitrary hom-bialgebra. Next we introduce the notion of \emph{hom-braided category} and show that this is the right setting for modules over quasitriangular hom-bialgebras. We also show how the hom-Yang-Baxter equation fits into this framework and how the category of Yetter-Drinfeld modules over a hom-bialgebra with bijective structure map can be organized as a hom-braided category. Finally we prove that, under certain conditions, one can obta…
Motivated by the universal obstruction to the deformation quantization of Poisson structures in infinite dimensions we introduce the notion of quantizable odd Lie bialgebra. The main result of the paper is a construction of a highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so called quantizable Poisson structures.
Let $H$ be a Hopf algebra with bijective antipode over a field $k$ and suppose that $R{#}H$ is a bi-product. Then $R$ is a bialgebra in the Yetter--Drinfel'd category ${}_H^H{\mathcal YD}$. We describe the bialgebras $(R{#}H)^{op}$ and $(R{#}H)^o$ explicitly as bi-products $R^{\UOP}{#}H^{op}$ and $R^{\UO}{#}H^o$ respectively where $R^{\UOP}$ is a bialgebra in ${}^{H^{op}}_{H^{op}}{\mathcal YD}$ and $R^{\UO}$ is a bialgebra in ${}^{H^o}_{H^o}{\mathcal YD}$. We use our results to describe two-cocycle twist bialgebra structures on the tensor product of bi-products.
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…
For any finite-dimensional Lie bialgebra $g$, we construct a bialgebra $A_{u,v}(g)$ over the ring $C[u][[v]]$, which quantizes simultaneously the universal enveloping bialgebra $U({g})$, the bialgebra dual to $U(g^*)$, and the symmetric bialgebra $S(g)$. We call $A_{u,v}(g)$ a biquantization of $S(g)$. We show that the bialgebra $A_{u,v}(g^*)$ quantizing $U(g^*)$, $U(g)^*$, and $S(g^*)$ is essentially dual to the bialgebra obtained from $A_{u,v}(g)$ by exchanging $u$ and $v$. Thus, $A_{u,v}(g)$ contains all information about the quantization of $g$. Our construction extends Etingof and Kazhdan…
In this paper, we associate a dual quasi-bialgebra, called bosonization, to every dual quasi-bialgebra $H$ and every bialgebra $R$ in the category of Yetter-Drinfeld modules over $H$. Then, using the fundamental theorem, we characterize as bosonizations the dual quasi-bialgebras with a projection onto a dual quasi-bialgebra with a preantipode. As an application we investigate the structure of the graded coalgebra $grA$ associated to a dual quasi-bialgebra $A$ with the dual Chevalley property (e.g. $A$ is pointed).
The aim of this paper is to establish a contravariant adjunction between the category of quasi-bialgebras and a suitable full subcategory of dual quasi-bialgebras, adapting the notion of finite dual to this framework. Various functorial constructions involving non-associative algebras and non-coassociative coalgebras are then carried out. Several examples illustrating our methods are expounded as well.
We propose a variant to the Etingof-Kazhdan construction of quantization functors. We construct the twistor J_Φassociated to an associator Φusing cohomological techniques. We then introduce a criterion ensuring that the ``left Hopf algebra'' of a quasitriangular QUE algebra is flat. We prove that this criterion is satisfied at the universal level. This provides a construction of quantization functors, equivalent to the Etingof-Kazhdan construction.
Any quantization of a quasitriangular Lie bialgebra g gives rise to a braiding of the dual Poisson-Lie formal group G^*. We show that this braiding always coincides with the Weinstein-Xu braiding. We also define the lifts of the classical r-matrix r as certain functions on G^* x G^*, prove their existence and uniqueness using co-Hochschild cohomology arguments and show that the lift can be expressed in terms of r by universal formulas.
The Lie bialgebras of the (1+1) extended Galilei algebra are obtained and classified into four multiparametric families. Their quantum deformations are obtained, together with the corresponding deformed Casimir operators. For the coboundary cases quantum universal R-matrices are also given. Applications of the quantum extended Galilei algebras to classical integrable systems are explicitly developed.
Let $L$ be a quantum semigroupoid, more precisely a $\times_R$-bialgebra in the sense of Takeuchi. We describe a procedure replacing the algebra $R$ by any Morita equivalent, or in fact more generally any $\sqrt{\text{Morita}}$ equivalent (in the sense of Takeuchi) algebra $S$ to obtain a $\times_S$-bialgebra $\widetilde H$ with the same monoidal representation category.
In the paper "On some unsolved problems in quantum group theory", V.Drinfeld formulated the problem of the existence of a universal quantization for Lie bialgebras. When the paper "Tensor structures arising from affine Lie algebras, III", by Kazhdan and Lusztig, appeared, Drinfeld asked whether its methods could be useful for the problem of universal quantization of Lie bialgebras. In this paper we use these methods to construct the universal quantization, which gives a positive answer to Drinfeld's question. We also show the existence of universal quantization of classical r-matrices, unitary…
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).
We prove that both, the embedding of the category of Hopf algebras into that of bialgebras and the forgetful functor from the category of Hopf algebras to the category of algebras, have right adjoints; in other words: every bialgebra has a Hopf coreflection and on every algebra there exists a cofree Hopf algebra. In this way we give an affirmative answer to a forty years old problem posed by Sweedler. On the route the coequalizers and the coproducts in the category of Hopf algebras are explicitly described.
The main aim of this paper is to investigate the structure of primitively generated connected braided bialgebras $A$ with respect to the braided vector space $P$ consisting of their primitive elements. When the Nichols algebra of $P$ is obtained dividing out the tensor algebra $T(P) $ by the two-sided ideal generated by its primitive elements of degree at least two, we show that $A$ can be recovered as a sort of universal enveloping algebra of $P$. One of the main applications of our construction is the description, in terms of universal enveloping algebras, of connected braided bialgebras who…
We introduce the analogue of Manin's universal coacting (bialgebra) Hopf algebra for Poisson algebras. First, for two given Poisson algebras $P$ and $U$, where $U$ is finite dimensional, we construct a Poisson algebra $\mathcal{B}(P,\, U)$ together with a Poisson algebra homomorphism $ψ_{\mathcal{B}(P,\,U)} \colon P \to U \otimes \mathcal{B}(P,\, U)$ satisfying a suitable universal property. $\mathcal{B}(P,\, U)$ is shown to admit a Poisson bialgebra structure for any pair of Poisson algebra homomorphisms subject to certain compatibility conditions. If $P=U$ is a finite dimensional Poisson alg…
In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras and cocommutative Hopf dialgebras. We also construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra and compare to the existing constructions. We are motivated by a differential geometric procedure which we call the Serre functor: To a pointed differentible manifold with multiplication is associated its distribution spa…
We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of the double. We show that there is a natural, but non-obvious way to construct a pre-Lie algebra from an arbitrary infinitesimal bialgebra and a dendriform algebra from a quasitriangular infinitesimal bialgebra. As consequences, we obtain a pre-Lie structure on the space of paths on an arbitrary quiver, and a striking dendriform structure on the space of endomorphisms of an arbitrary infinitesimal bialgeb…
Proposing a certain category of bialgebroid maps we show that the balanced depth 2 extensions appear as they were the finitary Galois extensions in the context of quantum groupoid actions, i.e., actions by finite bialgebroids, weak bialgebras or weak Hopf algebras. We comment on deformation of weak bialgebras, on half grouplike elements, on uniqueness of weak Hopf algebra reconstructions and discuss the example of separable field extensions.
De Concini, Kac, and Procesi defined a family of subalgebras Uq[w] of the quantized enveloping algebra Uq(g) associated to elements w in the Weyl group of a simple Lie algebra g. These algebras are called quantum Schubert cell algebras. We show that, up to a mild cocycle twist, quotients of certain quantum Schubert cell algebras of types E_6 and E_6^{(1)} map isomorphically onto distinguished subalgebras of the Faddeev-Reshetikhin-Takhtajan universal bialgebra associated to the braiding on the quantum half-spin representation of Uq(so_{10}). We identify the quotients as those obtained by facto…
To any field K of characteristic 0, we associate a set Sha(K). Elements of Sha(K) are equivalence classes of families of Lie polynomials subject to associativity relations. We construct an injection and a retraction between Sha(K) and the set of quantization functors of Lie bialgebras over K. This construction involves the following steps. 1) To each element \varpi of Sha(K), we associate a functor g\mapsto Sh(g) from the category of Lie algebras to that of Hopf algebras; Sh(g) contains Ug. 2) When g and h are Lie algebras, and r_{gh} \in g\otimes h, we construct an element R(r_{gh}) of Sh(g)\…
In the paper we introduce the notion of twisted derivation of a bialgebra. Twisted derivations appear as infinitesimal symmetries of the category of representations. More precisely they are infinitesimal versions of twisted automorphisms of bialgebras. Twisted derivations naturally form a Lie algebra (the tangent algebra of the group of twisted automorphisms). Moreover this Lie algebra fits into a crossed module (tangent to the crossed module of twisted automorphisms). Here we calculate this crossed module for universal enveloping algebras and for the Sweedler's Hopf algebra.
We establish a new and surprisingly strong link between two previously unrelated theories: the theory of moduli spaces of curves ${\mathcal M}_{g,n}$ (which, according to Penner, is controlled by the ribbon graph complex) and the homotopy theory of $E_d$ operads (controlled by ordinary graph complexes with no ribbon structure, introduced first by Kontsevich). The link between the two goes through a new intermediate {\em stable}\, ribbon graph complex which has roots in the deformation theory of quantum $A_\infty$ algebras and the theory of Kontsevich compactifications of moduli spaces of curve…
A braided bialgebra is called primitively generated if it is generated as an algebra by its space of primitive elements. We prove that any primitively generated braided bialgebra is isomorphic to the universal enveloping algebra of its infinitesimal braided Lie algebra, notions hereby introduced. This result can be regarded as a Milnor-Moore type theorem for primitively generated braided bialgebras and leads to the introduction of a concept of braided Lie algebra for an arbitrary braided vector space.
We introduce a general theory of twisting algebraic structures based on actions of a bialgebra. These twists are closely related to algebraic deformations and also to the theory of quasi-triangular bialgebras. In particular, a deformation produced from a universal deformation formula (UDF) is a special case of a twist. The most familiar example of a deformation produced from a UDF is perhaps the "Moyal product" which (locally) is the canonical quantization of the algebra of functions on a symplectic manifold in the direction of the Poisson bracket. In this case, the derivations comprising the …
Let $M_{*}({\bf C})$ denote the C$^{*}$-algebra defined as the direct sum of all matrix algebras $\{M_{n}({\bf C}):n\geq 1\}$. It is known that $M_{*}({\bf C})$ has a non-cocommutative comultiplication $Δ_φ$. We show that the C$^{*}$-bialgebra $(M_{*}({\bf C}),Δ_φ)$ has a universal $R$-matrix $R$ such that the quasi-cocommutative C$^{*}$-bialgebra $(M_{*}({\bf C}),Δ_φ,R)$ is triangular.
