Operad Structures Research Articles

On Schur algebras and derivations of free Lie algebras

We investigate the action of Schur algebra on the Lie algebras of derivations of free Lie algebras and operad structures constructed from it. We also show that the Lie algebra of derivations is generated by quadratic derivations together with the action of the Schur operad. Applications to certain subgroups of the automorphism group of a finitely generated free group are given as well.

Cliff operads: a hierarchy of operads on words

A new hierarchy of operads over the linear spans of $\delta$-cliffs, which are some words of integers, is introduced. These operads are intended to be analogues of the operad of permutations, also known as the associative symmetric operad. We obtain operads whose partial compositions can be described in terms of intervals of the lattice of $\delta$-cliffs. These operads are very peculiar in the world of the combinatorial operads since, despite to the relative simplicity for their construction, they are infinitely generated and they have nonquadratic and nonhomogeneous nontrivial relations. We provide a general construction for some of their quotients. We use it to endow the spaces of permutations, $m$-increasing trees, $c$-rectangular paths, and $m$-Dyck paths with operad structures. The operads on $c$-rectangular paths admit, as Koszul duals, operads generalizing the duplicial and triplicial operads.

Operad structures in geometric quantization of the moduli space of spatial polygons

The moduli space of spatial polygons is known as a symplectic manifold equipped with both Kähler and real polarizations. In this paper, associated to the Kähler and real polarizations, morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ are constructed by using the quantum Hilbert spaces $\mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}$ and $\mathscr{H}_\mathrm{re}$, respectively. Moreover, the relationship between the two morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ is studied and then the equality $\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}} = \dim \mathscr{H}_\mathrm{re}$ is proved in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama (2000) for proving $\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}} = \dim \mathscr{H}_\mathrm{re}$ in a special case.

Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations

We establish and explore a relationship between two approaches to moment-cumulant relations in free probability theory: on one side the main approach, due to Speicher, given in terms of Möbius inversion on the lattice of noncrossing partitions, and on the other side the more recent non-commutative shuffle-algebra approach, where the moment-cumulant relations take the form of certain exponential-logarithm relations. We achieve this by exhibiting two operad structures on (noncrossing) partitions, different in nature: one is an ordinary, non-symmetric operad whose composition law is given by insertion into gaps between elements, the other is a coloured, symmetric operad with composition law expressing refinement of blocks. We show that these operad structures interact so as to make the corresponding incidence bialgebra of the former a comodule bialgebra for the latter. Furthermore, this interaction is compatible with the shuffle structure and thus unveils how the two approaches are intertwined. Moreover, the constructions and results are general enough to extend to ordinary set partitions.

Operads of Finite Posets

We describe four natural operad structures on the linear species of finite posets. The three last ones are set-theoretical and can be seen as a simplified version of the first, the same way the NAP operad behaves with respect to the pre-Lie operad. Moreover the two first ones are isomorphic.

A proof of the existence of Batanin's initial operad

In this paper we prove the existence of the n-globular operad used in Batanin's definition of weak n-category. This operad is initial in the category of n-globular operads equipped with two extra pieces of structure: a system of compositions and a contraction. Our approach closely follows a proof by Leinster of the existence of a similar n-globular operad used in his definition of weak n-category (itself a variant of Batanin's definition) – we show that there is a functor giving the free operad equipped with a contraction and system of compositions on an n-globular collection, and applying this functor to the initial collection gives the desired initial operad. Since there is no interaction between the contraction and operad structures we are able to treat their free constructions separately. This is not true of the system of compositions structure, which cannot exist separately from the operad structure, so we use an interleaving-style construction to describe the free operad with system of compositions.

Reedy categories which encode the notion of category actions

We study a certain type of action of categories on categories and on operads. Using the structure of the categories {\Delta} and {\Omega} governing category and operad structures, respectively, we define categories which instead encode the structure of a category acting on a category, or a category acting on an operad. We prove that the former has the structure of an elegant Reedy category, whereas the latter has the structure of a generalized Reedy category. In particular, this approach gives a new way to regard group actions on categories and on operads.

Commutative ring objects in pro-categories and generalized Moore spectra

We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of M. J. Hopkins that certain towers of generalized Moore spectra, closely related to the K(n)-local sphere, are E-infinity algebras in the category of pro-spectra. In addition, we show that Adams resolutions automatically satisfy the above rigidity criterion. In order to carry this out we develop the concept of an operadic model category, whose objects have homotopically tractable endomorphism operads.

The operad of finite labeled lattices

This paper is a continuation of the series of papers [1–5]. We study various classes of finite labeled graphs (nonoriented or oriented) which form operads with respect to the composition operations introduced in [1]. The partially ordered sets and, in particular, lattices, form important classes of oriented graphs. The families of finite labeled partially ordered sets and lattices also possess operad structures. In this paper, we study elements of the operad of lattices that admit no nontrivial decomposition into operad composition. It turns out that the question on such lattices is closely related to the question on the structure of the congruence Θ(a, b) from [6]. Namely, an element of the operad of finite labeled lattices (i.e., a finite lattice) is operad indecomposable if and only if, on this lattice, there exist no congruences of the following form. Elements a ≤ b are fixed (where either a = 0 or b = 1), and x ∼ y if (and only if) either a ≤ x, y ≤ b or x = y. We succeeded in finding many examples of lattices with this property. Note that the operad composition on the family of finite partially ordered sets bears similarities to Plonka’s Agassiz sums (see the definition in [7], P. 46; [8], P. 314–315). It should be also noted that in the literature on the graph theory (independently of the literature on the universal algebra) one can find constructions similar to the operad composition from [1] (see, e.g., [9, 10]). The operad language, however, is not used. In this paper, we construct the operad of finite labeled lattices and its generalization, the operad of finite labeled partially ordered sets. We establish a criterion of operad decomposability for lattices. Using this criterion, we prove operad indecomposability of a series of families of lattices. In particular, the following classes of lattice turn out to be indecomposable: simple lattices, lattices decomposable into product, lattices with relative complements, modular lattices with complements, many of finite distributive lattices, and all finite geometric lattices. We use definitions, results, and notation from [1] with changes appeared later in [11]. Some preliminary versions of results of this paper were announced in [3–5]. Denote by Lat(n) the set of finite lattices with n elements enumerated by the numbers from 1 to n. We consider the same lattice with different enumerations of elements as different elements of Lat(n). The groupΣn of permutations of degree n acts on Lat(n) by permutation of numbers of elements. From the aforesaid it follows that the orbits of the action of Σn correspond to different lattices. We introduce an operad structure on the family Lat = {Lat(n) | n ≥ 1} regarding lattices as a partial case of oriented graphs: x ≤ y means that there is a unique arc from a vertex x to a vertex y. Then the definition of the composition for oriented graphs (with regard to the structure of partially ordered sets) takes the form

Classification of braids which give rise to interchange

It is well known that the existence of a braiding in a monoidal category V allows many structures to be built upon that foundation. These include a monoidal 2-category V-Cat of enriched categories and functors over V, a monoidal bicategory V-Mod of enriched categories and modules, a category of operads in V and a 2-fold monoidal category structure on V. We will begin by focusing our exposition on the first and last in this list due to their ability to shed light on a new question. We ask, given a braiding on V, what non-equal structures of a given kind in the list exist which are based upon the braiding. For instance, what non-equal monoidal structures are available on V-Cat, or what non-equal operad structures are available which base their associative structure on the braiding in V. We demonstrate alternative underlying braids that result in an infinite family of associative structures. The external and internal associativity diagrams in the axioms of a 2-fold monoidal category will provide us with several obstructions that can prevent a braid from underlying an associative structure.

681 The FDG standardized uptake value in predicting the outcome in head and neck cancer patients

In this paper we prove the existence of the n-globular operad used in Batanin's definition of weak n-category. This operad is initial in the category of n-globular operads equipped with two extra pieces of structure: a system of compositions and a contraction. Our approach closely follows a proof by Leinster of the existence of a similar n-globular operad used in his definition of weak n-category (itself a variant of Batanin's definition) – we show that there is a functor giving the free operad equipped with a contraction and system of compositions on an n-globular collection, and applying this functor to the initial collection gives the desired initial operad. Since there is no interaction between the contraction and operad structures we are able to treat their free constructions separately. This is not true of the system of compositions structure, which cannot exist separately from the operad structure, so we use an interleaving-style construction to describe the free operad with system of compositions.

Arc operads and arc algebras

Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identified with open subsets of a compactification due to Penner of a space closely related to Riemann's moduli space. Algebras over these operads are shown to be Batalin-Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy, and it similarly acts on the loop space of any topological space. New operad structures on the circle are classified and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.