The Category and Operad of Matching Dialgebras

This paper gives a systematic study of matching dialgebras corresponding to the operad As ⁽²⁾ in Zinbiel (2012) as the only Koszul self dual operad there other than the operads of associative algebras and Poisson algebras. The close relationship of matching dialgebras with semi-homomorphisms and matched pairs of associative algebras are established. By anti-symmetrizing, matching dialgerbas are also shown to give compatible Lie algebras, pre-Lie algebras and PostLie algebras. By the rewriting method, the operad of matching dialgebras is shown to be Koszul and the free objects are constructed in terms of tensor algebras. The operadic complex computing the homology of the matching dialgebras is made explicit.     

Coassociative magmatic bialgebras and the Fine numbers

We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n−2 operations of arity n. The dimension of the space of all the n-ary operations of this primitive operad turns out to be the Fine number F _n−1. In short, the triple of operads (As, Mag, MagFine) is good. The third author work is partially supported by FONDECYT Project 1060224     

Pluriassociative algebras II: The polydendriform operad and related operads

Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the dendriform operad, the Koszul dual of the diassociative operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of dendriform algebras, called γ-polydendriform algebras, so that 1-polydendriform algebras are dendriform algebras. For that, we consider the operads obtained as the Koszul duals of the γ-pluriassociative operads introduced by the author in a previous work. In the same manner as dendriform algebras are suitable devices to split associative operations into two parts, γ-polydendriform algebras seem adapted structures to split associative operations into 2γ operation so that some partial sums of these operations are associative. We provide a complete study of the γ-polydendriform operads, the underlying operads of the category of γ-polydendriform algebras. We exhibit several presentations by generators and relations, compute their Hilbert series, and construct free objects in the corresponding categories. We also provide consistent generalizations on a nonnegative integer parameter of the duplicial, triassociative and tridendriform operads, and of some operads of the operadic butterfly.     

Pluriassociative algebras I: The pluriassociative operad

Diassociative algebras form a category of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of diassociative algebras, called γ-pluriassociative algebras, so that 1-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with 2γ associative binary operations satisfying some relations. We provide a complete study of the γ-pluriassociative operads, the underlying operads of the category of γ-pluriassociative algebras. We exhibit a realization of these operads, establish several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct the free objects in the corresponding categories. We also study several notions of units in γ-pluriassociative algebras and propose a general way to construct such algebras. This paper ends with the introduction of an analogous generalization of the triassociative operad of Loday and Ronco.     

La construction bar d'une algèbre comme algèbre de Hopf E-infini

We prove that the bar construction of an E_∞ algebra forms an E_∞ algebra. To be more precise, we provide the bar construction of an algebra over the surjection operad with the structure of a Hopf algebra over the Barratt–Eccles operad. (The surjection operad and the Barratt–Eccles operad are classical E_∞ operads.) To cite this article: B. Fresse, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On the algebra of quasi-shuffles

For any commutative algebra R the shuffle product on the tensor module T(R) can be deformed to a new product. It is called the quasi-shuffle algebra, or stuffle algebra, and denoted T ^q (R). We show that if R is the polynomial algebra, then T ^q (R) is free for some algebraic structure called Commutative TriDendriform (CTD-algebras). This result is part of a structure theorem for CTD-bialgebras which are associative as coalgebras and whose primitive part is commutative. In other words, there is a good triple of operads (As, CTD, Com) analogous to (Com, As, Lie). In the last part we give a similar interpretation of the quasi-shuffle algebra in the noncommutative setting.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

On the operad of bigraft algebras

In this paper, we study the notion of a bigraft algebra, generalizing the notions of left and right graft algebras. We construct the free bigraft algebra on one generator in terms of certain planar rooted trees with decorated edges, and therefore describe explicitly the bigraft operad. We then compute its Koszul dual and show that the bigraft operad is Koszul. Moreover, we endow the free bigraft algebra on one generator with a universal Hopf algebra structure and a pairing. Finally, we prove an analogue of the Poincaré–Birkhoff–Witt and Cartier–Milnor–Moore theorems. For this, we define the notion of infinitesimal bigraft bialgebras and we prove the existence of a new good triple of operads.     

Symmetrisation of n-operads and compactification of real configuration spaces

It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint Sym₁ given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author's theory of higher operads, the nonsymmetric operads are 1-operads and Sym₁ is the first term of the infinite series of left adjoint functors Symn, called symmetrisation functors, from n-operads to symmetric operads with the property that the category of one object, one arrow, …, one (n−1)-arrow algebras of an n-operad A is isomorphic to the category of algebras of Symn(A).In this paper we consider some geometrical and homotopical aspects of the symmetrisation of n-operads. We follow Getzler and Jones and consider their decomposition of the Fulton-Macpherson operad of compactified real configuration spaces. We construct an n-operadic counterpart of this compactification which we call the Getzler-Jones operad. We study the properties of Getzler-Jones operad and find that it is contractible and cofibrant in an appropriate model category. The symmetrisation of the Getzler-Jones operad turns out to be exactly the operad of Fulton and Macpherson. These results should be considered as an extension of Stasheff's theory of 1-fold loop spaces to n-fold loop spaces n?2. We also show that a space X with an action of a contractible n-operad has a natural structure of an algebra over an operad weakly equivalent to the little n-disks operad. A similar result holds for chain operads. These results generalise the classical Eckman-Hilton argument to arbitrary dimension.Finally, we apply the techniques to the Swiss-Cheese type operads introduced by Voronov and prove analogous results in this case.     

Up-to-homotopy algebras with strict units

We show that an operad with non-trivial arity zero admits a minimal model in the sense of Sullivan. Hence an up-to-homotopy algebra with a strict unit is just an operad algebra over such a minimal model. We also establish the descent of formality for certain unitary operads. As an application, we give another proof of the formality of the unitary n-little disks operad over the rationals.     

On Hopf algebra structures over free operads

The operad Lie can be constructed as the operad of primitives PrimAs from the operad As of associative algebras. This is reflected by the theorems of Friedrichs, Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore. We replace the operad As by families of free operads P, which include the operad Mag freely generated by a non-commutative non-associative binary operation and the operad of Stasheff polytopes. We obtain Poincaré-Birkhoff-Witt type theorems and collect information about the operads PrimP, e.g., in terms of characteristic functions.     

Solvability of universal bol 2-loops

ABSTRACT

Let (A, ^?) be a structurable algebra. Then the opposite algebra (A ^op , ^?) is structurable, and we show that the triple system B ^op _A(x, y, z):=V^opx,y(z)=x(y¯z)+z(y¯x)?y(x¯z), x, y, z ∈ A, is a Kantor triple system (or generalized Jordan triple system of the second order) satisfying the condition (A). Furthermore, if A=𝔸₁?𝔸₂ denotes tensor products of composition algebras, (^?) is the standard conjugation, and (^∧) denotes a certain pseudoconjugation on A, we show that the triple systems B ^op _𝔸1?𝔸2 ( x , y¯^∧, z) are models of compact Kantor triple systems. Moreover these triple systems are simple if (dim𝔸₁, dim𝔸₂) ≠ (2, 2). In addition, we obtain an explicit formula for the canonical trace form for compact Kantor triple systems defined on tensor products of composition algebras.     

Adem-Cartan Operads

In this article, we introduce Adem-Cartan operads and prove that the cohomology of any algebra over such an operad is an unstable level algebra over the extended Steenrod algebra. Moreover, we prove that this cohomology is endowed with secondary cohomology operations.     

Operads,configuration spaces and quantization

We review several well-known operads of compactified configuration spaces and construct several new such operads, [`(C)]\bar C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A <sub>∞</sub>-algebras and their homotopy morphisms, (ii) the 2-coloured operad of L <sub>∞</sub>-algebras and their homotopy morphisms, and (iii) the 4-coloured operad of openclosed homotopy algebras and their homotopy morphisms. Two gadgets — a (coloured) operad of Feynman graphs and a de Rham field theory on [`(C)]\bar C — are introduced and used to construct quantized representations of the (fundamental) chain operad of [`(C)]\bar C which are given by Feynman type sums over graphs and depend on choices of propagators.     

Operads within Monoidal Pseudo Algebras

A general notion of operad is given, which includes: (1) the operads that arose in algebraic topology in the 1970s to characterise loop spaces. (2) the higher operads of Michael Batanin [4] (3) braided and symmetric analogues of Batanin’s operads which are likely to be important in the study of weakly symmetric higher dimensional monoidal categories. The framework of this paper, links together two-dimensional monad theory, operads, and higher dimensional algebra, in a natural way.     

A Generalized Drinfeld Quantum Double Construction Based on Weak Hopf Algebras

Let B and H be weak Hopf algebras with bijective antipodes S _B and S _H , respectively. Based on a compatible weak Hopf dual pairing (B, H, σ), we construct a generalized Drinfeld quantum double 𝔻(B, H) which is a weak T-coalgebra over a twisted semi-direct square of groups. In particular, when B and H are finite dimensional and the above pairing map σ is nondegenerate, 𝔻(B, H) admits a nontrivial quasitriangular structure. Some explicit examples are given as an application of our theory.     

Self-dual hopf algebras

We introduce the notions of self-dual (graded) Hopf algebras and of structurally simple (graded) Hopf algebras. We prove that the self-dual Hopf algebras are structurally simple and provide a construction of self-dual Hopf algebras. Finally, we classify the self-dual quotients of the form T_B (M)/I, where T_B (M) is a path algebra with a graded Hopf algebra structure, and I is a graded admissible Hopf ideal.