We introduce a concept of $\frac23$PROP generalizing the Kontsevich concept of $\frac12$PROP. We prove that some Stasheff-type compactification of the Kontsevich spaces $K(m,n)$ defines a topological $\frac23$PROP structure. The corresponding chain complex is a minimal model for its cohomology (both are considered as $\frac23$PROPs). We construct a $\frac23$PROP $\End(V)$ for a vector space $V$. Finally, we construct a dg Lie algebra controlling the deformations of a (co)associative bialgebra. Philosophically, this construction is a version of the Markl's operadic construction from [M1] applied to minimal models of $\frac23$PROPs.
We compactify the spaces $K(m,n)$ introduced by Maxim Kontsevich. The initial idea was to construct an $L_\infty$ algebra governing the deformations of a (co)associative bialgebra. However, this compactification leads not to a resolution of the PROP of (co)associative bialgebras, but to a new algebraic structure we call here a CROC. It turns out that these constructions are related to the non-commutative deformations of (co)associative bialgebras. We construct an associative dg algebra conjecturally governing the non-commutative deformations of a bialgebra. Then, using the Quillen duality, we …
This work is devoted to study new bialgebra structures related to 2-associative algebras. A 2-associative algebra is a vector space equipped with two associative multiplications. We discuss the notions of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras. The first structure was revealed by J.-L. Loday and M. Ronco in an analogue of a Cartier-Milnor-Moore theorem, the second was suggested by Loday and the third is a variation of the second one. The main results of this paper are the construction of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras starting from an associative a…
Basic definitions and properties of nearly associative algebras are described. Nearly associative algebras are proved to be Lie-admissible algebras. Two-dimensional nearly associative algebras are classified, and its main classes are derived. The bimodules, matched pairs and Manin triple of a nearly associative algebras are derived and their equivalence with nearly associative bialgebras is proved. Basic definitions and properties of nearly Hom-associative algebras are described. Related bimodules and matched pairs are given, and associated identities are established.
We introduce the concept of {sigma, tau}-Rota-Baxter operator, as a twisted version of a Rota-Baxter operator of weight zero. We show how to obtain a certain {sigma, tau}-Rota-Baxter operator from a solution of the associative (Bi)Hom-Yang-Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.
We establish a dictionary between the Cacti algebra axioms on a Cacti algebra structure with underlying free associative algebra, under suitable good behavior with degrees. Using these ideas, for an associative algebra $A$ and a bialgebra $H$, we also translate Cacti algebra maps $Ω(H)\to C^\bullet(A)$ (where $Ω(H)$ stands for the cobar construction on $H$ and $C^\bullet(A)$ is the Hochschild cohomology complex) with $H$-module algebra structures on $A$, and illustrate with examples of applications.
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.
A BiHom-associative algebra is a (nonassociative) algebra $A$ endowed with two commuting multiplicative linear maps $α,β\colon A\rightarrow A$ such that $α(a)(bc)=(ab)β(c)$, for all $a, b, c\in A$. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).
Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the context of Rota-Baxter algebras. The study of the coboundary case leads to an enrichment of the associative Yang-B…
In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as $\mathbb{C}[\partial]$-modules such that both of them are subalgebras an…
For a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra L, let T(L) be the vector space of tensors over L equipped with the Ito Hopf algebra structure derived from the associative multiplication in L. We show a necessary and sufficient condition that the double product integral satisfy the quantum Yang-Baxter equation over T(L). We construct a quantisation of an arbitrary quasitriangular Lie bialgebra structure on L in the unital associative subalgebra of T(L)[[h]] consisting of formal power series whose zero order coefficient lies in the space S(L) of s…
We study Hom-bialgebras and objects admitting coactions by Hom-bialgebras. In particular, we construct a Hom-bialgebra M representing the functor of 2x2-matrices on Hom-associative algebras. Then we construct a Hom-algebra analogue of the affine plane and show that it is a comodule Hom-algebra over M in a suitable sense. It is also shown that the enveloping Hom-associative algebra of a Hom-Lie algebra is naturally a Hom-bialgebra.
A generalisation of the notion of a Rota-Baxter operator is proposed. This generalisation consists of two operators acting on an associative algebra and satisfying equations similar to the Rota-Baxter equation. Rota-Baxter operators of any weights and twisted Rota-Baxter operators are solutions of the proposed system. It is shown that dendriform algebra structures of a particular kind are equivalent to Rota-Baxter systems. It is shown further that a Rota-Baxter system induces a weak peudotwistor [F. Panaite & F. Van Oystaeyen, Twisted algebras, twisted bialgebras and Rota-Baxter operators, arX…
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 construct an associative algebra with a decomposition into the direct sum of the underlying vector spaces of another associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is a Connes cocycle. The former is called a double construction of Frobenius algebra and the latter is called a double construction of Connes cocycle which is interpreted in terms of dendriform algebras. Both of them are equivalent to a kind of bialgebras, namely, antisymmetric infinitesimal bialgebras an…
We define a graded multiplication on the vector space of essential paths on a graph $G$ (a tree) and show that it is associative. In most interesting applications, this tree is an ADE Dynkin diagram. The vector space of length preserving endomorphisms of essential paths has a grading obtained from the length of paths and possesses several interesting bialgebra structures. One of these, the Double Triangle Algebra (DTA) of A. Ocneanu, is a particular kind of quantum groupoid (a weak Hopf algebra) and was studied elsewhere; its coproduct gives a filtrated convolution product on the dual vector s…
The vector space spanned by rooted forests admits two graded bialgebra structures. The first is defined by A. Connes and D. Kreimer using admissible cuts, and the second is defined by D. Calaque, K. Ebrahimi-Fard and the second author using contraction of trees. In this article we define the doubling of these two spaces. We construct two bialgebra structures on these spaces which are in interaction, as well as two related associative products obtained by dualization. We also show that these two bialgebras verify a commutative diagram similar to the diagram verified D. Calaque, K. Ebrahimi-Fard…
The notion of PROP models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. We prove a Koszul duality theory for PROPs generalizing the one for associative algebras and for operads.
The notion of prop models the operations with multiple inputs and multiple outpus, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props.
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…
We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality t…
We compute the homotopy derivations of the properads governing even and odd Lie bialgebras as well as involutive Lie bialgebras. The answer may be expressed in terms of the Kontsevich graph complexes. In particular, this shows that the Grothendieck-Teichmüller group acts faithfully (and essentially transitively) on the completions of the properads governing even Lie bialgebras and involutive Lie bialgebras, up to homotopy. This shows also that by contrast to the even case the properad governing odd Lie bialgebras admits precisely one non-trivial automorphism - the standard rescaling automorphi…
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.
For any integer $d$ we introduce a prop $RHra_d$ of oriented ribbon hypergraphs (in which "edges" can connect more than two vertices) and prove that it admits a canonical morphism of props, $$ Holieb_d^\diamond \longrightarrow RHra_d, $$ $Holieb_d^\diamond$ being the (degree shifted) minimal resolution of prop of involutive Lie bialgebras, which is non-trivial on every generator of $Holieb_d^\diamond$. We obtain two applications of this general construction. As a first application we show that for any graded vector space $W$ equipped with a family of cyclically (skew)symmetric higher products …
We show Koszulness of the prop governing involutive Lie bialgebras and also of the props governing non-unital and unital-counital Frobenius algebras, solving a long-standing problem. This gives us minimal models for their deformation complexes, and for deformation complexes of their algebras which are discussed in detail. Using an operad of graph complexes we prove, with the help of an earlier result of one of the authors, that there is a highly non-trivial action of the Grothendieck-Teichmüller group $GRT_1$ on (completed versions of) the minimal models of the properads governing Lie bialgebr…
We analyze among all possible quantum deformations of the 3+1 (anti)de Sitter algebras, so(3,2) and so(4,1), which have two specific non-deformed or primitive commuting operators: the time translation/energy generator and a rotation. We prove that under these conditions there are only two families of two-parametric (anti)de Sitter Lie bialgebras. All the deformation parameters appearing in the bialgebras are dimensionful ones and they may be related to the Planck length. Some properties conveyed by the corresponding quantum deformations (zero-curvature and non-relativistic limits, space isotro…
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…
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 $H$.
