Related papers: Simplicial Structure on Connected Multiplicative O…
Motivated by the recent work of Batkam-Tcheka on pointed multiplicative operads, we construct in this paper new chain complex algebras and two distinct bicomplex algebra structures on a free symmetric connected multiplicative differential…
We introduce and study structured enhancement of the notion of a crossed simplicial group, which we call an operadic crossed simplicial group. We show that with each operadic crossed simplicial group one can associate a certain operad in…
We define a construction on operads which yields a new description of the minimal model. The construction also allows us to define algebraic structures on the homology of chain complexes with homologously trivial operad algebra structures,…
We introduce a systematic method for constructing set-theoretic operads via iterated application of the power set functor, and use it to uncover a hierarchy connecting several classical operads. Starting from the permutative operad, the…
The associative operad is a certain algebraic structure on the sequence of group algebras of the symmetric groups. The weak order is a partial order on the symmetric group. There is a natural linear basis of each symmetric group algebra,…
We introduce a simplicial object $(\{ \Dy^m\}_{m\geq 0}, {\mathbb F}_i, {\mathbb S}_j)$ in the category of non-symmetric algebraic operads, satisfying that $\Dy^0$ is the operad of associative algebras and $\Dy^1$ is J.-L. Loday\rq s operad…
A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor…
Given a symplectic manifold $M$, we may define an operad structure on the the spaces $\op^k$ of the Lagrangian submanifolds of $(\bar{M})^k\times M$ via symplectic reduction. If $M$ is also a symplectic groupoid, then its multiplication…
We generalize the construction of multitildes in the aim to provide multitilde operators for regular languages. We show that the underliying algebraic structure involves the action of some operads. An operad is an algebraic structure that…
We provide a general notion of induced structures of operated algebras in the context of unary-binary operads. This notion fully captures the binary quadratic relations encoded by a unary-binary operad, thereby unifying and formalizing the…
The main ideas developed in this habilitation thesis consist in endowing combinatorial objects (words, permutations, trees, Young tableaux, etc.) with operations in order to construct algebraic structures. This process allows, by studying…
We study different algebraic structures associated to an operad and their relations: to any operad $\mathbf{P}$ is attached a bialgebra,the monoid of characters of this bialgebra, the underlying pre-Lie algebra and its enveloping algebra;…
This paper establishes a uniform procedure to split the operations in any algebraic operad, generalizing previous known notions of splitting algebraic structures from the dendriform algebra of Loday that splits the associative operation to…
In this paper, we use the language of operads to study open dynamical systems. More specifically, we study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones. The syntactic architecture of…
We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many…
Noncommutative multi-indices are noncommutative monomials in a $\mathbb{N}$-indexed family of indeterminates. We define on them a $\mathbb{Z}$-graded operadic structure, with the help of a shifting derivation. Multi-indices of degree 0 are…
Modular operads relevant to string theory can be equipped with an additional structure, coming from the connected sum of surfaces. Motivated by this example, we introduce a notion of connected sum for general modular operads. We show that a…
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…
Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer…
We construct an analogue of the Livernet--Loday operad for two compatible brackets. The Livernet--Loday operad can be used to define $\star$-products and deformation quantization for Poisson structures. We make use of our operad in the same…