Related papers: Permutahedral Structures of $E_2$ Operads
This paper is built on the following observation: the purity of the mixed Hodge structure on the cohomology of Brown's moduli spaces is essentially equivalent to the freeness of the dihedral operad underlying the gravity operad. We prove…
In this paper we take up again the deformation theory for $K$-linear pseudofunctors initiated in a previous work (Adv. Math. 182 (2004) 204-277). We start by introducing a notion of a 2-cosemisimplicial object in an arbitrary 2-category and…
In this paper, we first give the notation of a compatible pre-Lie algebra and its representation. We study the relation between compatible Lie algebras and compatible pre-Lie algebras. We also construct a new bidifferential graded Lie…
Weighted Rota-Baxter operators on associative algebras are closely related to modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal bialgebras, and play an important role in mathematical physics. For any $\lambda \in…
We attack the question of E_2-formality of differential graded algebras over prime fields via obstruction theory. We are able to prove that E_2-algebras whose cohomology ring is a polynomial algebra on even degree classes are intrinsically…
We endow categories of non-symmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories…
We prove the conjectures on dimensions and characters of some quadratic algebras stated by B$.$L$.$Feigin. It turns out that these algebras are naturally isomorphic to the duals of the components of the bihamiltonian operad.
In this paper, we prove a conjecture of Blumberg and Hill regarding the existence of $N_\infty$-operads associated to given sequences $\mathcal{F} = (\mathcal{F}_n)_{n \in \mathbb{N}}$ of families of subgroups of $G\times \Sigma_n$. For…
We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration and…
For operads with a map from the curved homotopy Lie operad, we introduce a corresponding curved variant `cTw' of Willwacher's operadic twisting comonad `Tw'. We show that cTw-coalgebra structures on such an operad are in bijection with…
We present a theory of Dyer-Lashof operations in unoriented bordism (the canonical splitting $N_*(X)\simeq N_*\otimes H_*(X)$, where $N_*( )$ is unoriented bordism and $H_*( )$ is homology mod 2, does not respect these operations). For any…
We use the differential algebra of polytopes to explain the known remarkable relation of the combinatorics of the associahedra and permutohedra with the universal compositional and multiplicative inversion formulas for the formal power…
We study the Andr\'e-Quillen cohomology with coefficients of an algebra over an operad. Using resolutions of algebras coming from Koszul duality theory, we make this cohomology theory explicit and we give a Lie theoretic interpretation. For…
In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is different from the standard construction of a monad from an operad in that the…
Let $M^n$ be a compact orientable smooth Riemannian submanifold of dimension $n\geq 3$ in $\mathbb R^d$. We construct a family of deformed Hodge Laplacians $\Delta_t^*$, $t>0$, acting on differential forms and defined through the extrinsic…
We construct an enlargement of the classifying space of mixed Hodge structures with polarized graded quotients, by adding mixed Hodge theoretic version of SL(2)-orbits. This space has a real analytic structure and a log structure with sign.…
This paper is the first in a series of works devoted to an operadic study of Nijenhuis structures, focusing on Nijenhuis associative algebras. We introduce the concept of homotopy Nijenhuis associative algebras and demonstrate that the…
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…
We introduce unary operadic 2-categories as a framework for operadic Grothendieck construction for categorical $\mathbb{O}$-operads, $\mathbb{O}$ being a unary operadic category. The construction is a fully faithful functor…
The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the…