Associative Operad Research Articles

<i>A<sub>∞</sub></i>-algebra Structure on Connected Multiplicative Operad

This work develops the structure of <i>A<sub>∞</sub></i>-algebras on operad theory and also the preservation of this structure by a morphism of operads well defined. This structure defined here is motivated by the important role that play certain particular properties such as multiplication and connectivity on the operads. Another key ingredient used to develop this work is the brace operations; which, combined with the properties cited above allowed to better frame the study of this structure. Thus, this paper show explicitly the existence of an <i>A<sub>∞</sub></i>-algebra structure on any connected multiplicative operad endowed with its brace operations and that this structure is minimal if the operad is only multiplicative. Furthermore, the paper also shows the existence of an operads morphism from an unital associative operad, <i>A<sub>ss</sub></i> to any connected multiplicative operad 𝒪 preserving the structure of <i>A<sub>∞</sub></i>-algebras existing on these two operads. And when the operad 𝒪 is just multiplicative then there is rather a morphism of operads from the associative operad, Asto 𝒪 preserving this time the minimal <i>A<sub>∞</sub></i>-algebras structure existing on these operads.

Distributive laws between the operads Lie and Com

Using methods of computer algebra, especially, Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet–Loday formula deforming the Poisson operad into the associative operad.

Twisting structures and morphisms up to strong homotopy

We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the “strong homotopy” morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads , which is exactly the two-sided Koszul resolution of the associative operad , also known as the Alexander-Whitney co-ring.

Quotients of the Magmatic Operad: Lattice Structures and Convergent Rewrite Systems

We study quotients of the magmatic operad that is the free nonsymmetric operad over one binary generator. In the linear setting, we show that the set of these quotients admits a lattice structure and we show an analog of the Grassmann formula for the dimensions of these operads. In the nonlinear setting, we define comb associative operads that are operads indexed by non-negative integers generalizing the associative operad. We show that the set of comb associative operads admits a lattice structure, isomorphic to the lattice of non-negative integers equipped with the division order. Driven by computer experimentations, we provide a finite convergent presentation for the comb associative operad in correspondence with 3. Finally, we study quotients of the magmatic operad by one cubic relation by expressing their Hilbert series and providing combinatorial realizations.

Classification of Regular Parametrized One-relation Operads

AbstractJean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: . Such an operad is called a parametrized one-relation operad. For a particular choice of parameters {xσ}, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space V is, as a graded vector space, isomorphic to the tensor algebra of V. We classify, over an algebraically closed field of characteristic zero, all regular parametrized one-relation operads. In fact, we prove that each such operad is isomorphic to one of the following ûve operads: the left-nilpotent operad defined by the relation ((a1a2)a3) = 0, the associative operad, the Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our computationalmethods combine linear algebra over polynomial rings, representation theory of the symmetric group, and Gröbner bases for determinantal ideals and their radicals.

Symmetric multiplicative formality of the Kontsevich operad

In his famous paper entitled “Operads and motives in deformation quantization”, Maxim Kontsevich constructed (in order to prove the formality of the little d-disks operad) a topological operad, which is called in the literature the Kontsevich operad, and which is denoted $${\mathcal {K}}_d$$ in this paper. This operad has a nice structure: it is a multiplicative symmetric operad, that is, it comes with a morphism from the symmetric associative operad. There are many results in the literature regarding the formality of $${\mathcal {K}}_d$$ . It is well known (by Kontsevich) that $${\mathcal {K}}_d$$ is formal over reals as a symmetric operad. It is also well known (independently by Syunji Moriya and the author) that $${\mathcal {K}}_d$$ is formal as a multiplicative nonsymmetric operad. In this paper, we prove that the Kontsevich operad is formal over reals as a multiplicative symmetric operad, when $$d \ge 3$$ .

Modular envelopes, OSFT and nonsymmetric (non-$\Sigma$) modular operads

Our aim is to introduce and advocate non- \Sigma (non-symmetric) modular operads. While ordinary modular operads were inspired by the structure of the moduli space of stable complex curves, non- \Sigma modular operads model surfaces with open strings outputs. An immediate application of our theory is a short proof that the modular envelope of the associative operad is the linearization of the terminal operad in the category of non- \Sigma modular operads. This gives a succinct description of this object that plays an important rôle in open string field theory . We also sketch further perspectives of the presented approach.

One-parameter deformations of the diassociative and dendriform operads

Livernet and Loday constructed a polarization of the nonsymmetric associative operad with one operation into a symmetric operad with two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of , which includes Poisson algebras. We combine this with the dendriform splitting of an associative operation into the sum of two nonassociative operations, and use Koszul duality for quadratic operads, to construct one-parameter deformations of the nonsymmetric dendriform and diassociative operads into the category of symmetric operads.

Operads from posets and Koszul duality

We introduce a functor As from the category of posets to the category of nonsymmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction As provides a generalization of the associative operad because all of its generating operations are associative. This construction has a very singular property: the operads obtained from As are almost never basic. Besides, the properties of the obtained operads, such as Koszulity, basicity, associative elements, realization, and dimensions, depend on combinatorial properties of the starting posets. Among others, we show that the property of being a forest for the Hasse diagram of the starting poset implies that the obtained operad is Koszul. Moreover, we show that the construction As restricted to a certain family of posets with Hasse diagrams satisfying some combinatorial properties is closed under Koszul duality.

Homotopy units in 𝐴-infinity algebras

We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra structures on a given object is up to homotopy a subset of connected components of the space of non-unital associative algebra structures.

Delooping totalization of a multiplicative operad

The paper shows that under some conditions the totalization of a cosimplicial space obtained from a multiplicative operad is a double loop space of the space of derived morphisms from the associative operad to the operad itself.

Hairy graphs and the unstable homology of Mod(g , s ), Out(F n ) and Aut(F n )

We study a family of Lie algebras hO which are defined for cyclic operads O. Using his graph homology theory, Kontsevich identified the homology of two of these Lie algebras (corresponding to the Lie and associative operads) with the cohomology of outer automorphism groups of free groups and mapping class groups of punctured surfaces, respectively. In this paper, we introduce a hairy graph homology theory for O. We show that the homology of hO embeds in hairy graph homology via a trace map that generalizes the trace map defined by Morita. For the Lie operad, we use the trace map to find large new summands of the abelianization of hO which are related to classical modular forms for SL2(ℤ). Using cusp forms, we construct new cycles for the unstable homology of Out(Fn), and using Eisenstein series, we find new cycles for Aut(Fn). For the associative operad, we compute the first homology of the hairy graph complex by adapting an argument of Morita, Sakasai and Suzuki, who determined the complete abelianization of hO in the associative case.

Long knots and maps between operads

We identify the space of tangentially straightened long knots in R^m, for m greater than or equal to 4, as the double loops on the space of derived operad maps from the associative operad into a version of the little m-disk operad. This verifies a conjecture of Kontsevich, Lambrechts, and Turchin.

Koszul duality in algebraic topology

The most prevalent examples of Koszul duality of operads are the self-duality of the associative operad and the duality between the Lie and commutative operads. At the level of algebras and coalgebras, the former duality was first noticed as such by Moore, as announced in his ICM talk at Nice (Moore in Actes du Congres International des Mathematiciens, Tome 1. Gauthier-Villars, Paris, pp. 335–339, 1971). This particular duality has typically been called Moore duality, and some prefer to call the general phenomenon Koszul–Moore duality. The second duality at the level of algebras was realized in the seminal work of Quillen on rational homotopy theory (Quillen in Ann Math 90(2):205–295, 1969). Our aim in these notes based on our talk at the Luminy workshop on Operads in 2009 is to try to provide some historical, topological context for these two classical algebraic dualities. We first review the original cobar and bar constructions used to study loop spaces and classifying spaces, emphasizing the less-familiar geometry of the cobar construction. Then, after some elementary topology, we state duality between bar and cobar complexes in that setting. Before explaining Quillen’s work, we also share some other ideas—calculations of Cartan–Serre and Milnor–Moore and philosophy of Eckmann–Hilton—which may have influenced him. After stating Quillen’s duality, we share some recent work which relates these constructions to geometry through Hopf invariants and in particular linking phenomena.

Constructing combinatorial operads from monoids

We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

The framed little 2-discs operad and diffeomorphisms of handlebodies

The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (that is, the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-dimensional handlebodies with marked discs on their boundaries. A modification of the argument provides a new and elementary proof of Costello's theorem that the derived modular envelope of the associative operad is homotopy equivalent to the ‘open string’ modular operad made from moduli spaces of Riemann surfaces with marked intervals on the boundary. Our technique also recovers a theorem of Braun that the derived modular envelope of the cyclic operad that describes associative algebras with involution is homotopy equivalent to the modular operad made from moduli spaces of unoriented Klein surfaces with open string gluing.

On the Extension of a TCFT to the Boundary of the Moduli Space

The purpose of this paper is to describe an analogue of a construction of Costello in the context of finite-dimensional differential graded Frobenius algebras which produces closed forms on the decorated moduli space of Riemann surfaces. We show that this construction extends to a certain natural compactification of the moduli space which is associated to the modular closure of the associative operad, due to the absence of ultra-violet divergences in the finite-dimensional case. We demonstrate that this construction is equivalent to the "dual construction" of Kontsevich.

Koszul duality of En-operads

The goal of this paper is to prove a Koszul duality result for E_n-operads in differential graded modules over a ring. The case of an E_1-operad, which is equivalent to the associative operad, is classical. For n>1, the homology of an E_n-operad is identified with the n-Gerstenhaber operad and forms another well known Koszul operad. Our main theorem asserts that an operadic cobar construction on the dual cooperad of an E_n-operad defines a cofibrant model of E_n. This cofibrant model gives a realization at the chain level of the minimal model of the n-Gerstenhaber operad arising from Koszul duality. Most models of E_n-operads in differential graded modules come in nested sequences of operads homotopically equivalent to the sequence of the chain operads of little cubes. In our main theorem, we also define a model of the operad embeddings E_n-1 --> E_n at the level of cobar constructions.

On the non-Koszulity of ternary partially associative operad

We prove that the operad for ternary partially associative algebras is non Koszul. The aim is to underline the problem of computing the dual operad when we consider quadratic operad for n-ary algebras in particular when n is odd. In fact, the dual operad is generally defined in the graded (differential) operad framework. The result of non-Koszulity extends for other operads for (2p+ 1)-ary partially associative algebras although the operads for (2p)-ary partially associative algebras are Koszul.