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17
Relating Two Hopf Algebras Built from An Operad, by appear
"... Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morph ..."
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Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the NAP operad, where the incidence Hopf algebra is identified with the ConnesKreimer Hopf algebra of rooted trees. 1
On the algebra of quasishuffles
 Manuscripta Mathematica
"... Abstract. 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 quasishuffle 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 cal ..."
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Abstract. 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 quasishuffle 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 (CTDalgebras). This result is part of a structure theorem for CTDbialgebras which are associative as coalgebras and whose primitive part is commutative. In other words, there is a good triple of operads (As, CT D, Com) analogous to (Com, As, Lie). In the last part we give a similar interpretation of the quasishuffle algebra in the noncommutative setting.
MANIN PRODUCTS, KOSZUL DUALITY, LODAY ALGEBRAS AND DELIGNE CONJECTURE
"... Dedicated to JeanLouis Loday, on the occasion of his sixtieth birthday 1 Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, nonsymmetric operads, operads, ..."
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Dedicated to JeanLouis Loday, on the occasion of his sixtieth birthday 1 Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, nonsymmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these operations, we are able to prove that Deligne’s conjecture holds for a general class of operads and is not specific to the case of associative algebras. Finally, we prove generalized versions of a few conjectures raised by M. Aguiar and J.L. Loday related to the Koszul property of operads defined by black products. These operads provide infinitely many examples for this generalized Deligne’s conjecture.
Character formulas for the operad of two compatible brackets and for the bihamiltonian operad
"... Abstract. We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the SL2 group in these spaces. 1. ..."
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Abstract. We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the SL2 group in these spaces. 1.
Operads of compatible structures and weighted partitions
 J. Pure Appl. Algebra
"... Abstract. In this paper we describe operads encoding two different kinds of compatibility of algebraic structures. We show that there exist decompositions of these in terms of black and white products and we prove that they are Koszul for a large class of algebraic structures by using the poset meth ..."
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Abstract. In this paper we describe operads encoding two different kinds of compatibility of algebraic structures. We show that there exist decompositions of these in terms of black and white products and we prove that they are Koszul for a large class of algebraic structures by using the poset method of B. Vallette. In particular we show that this is true for the operads of compatible Lie, associative and preLie algebras.
Operad profiles of Nijenhuis structures
, 2008
"... Abstract. Recently S. Merkulov [Mer04, Mer05, Mer06] established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as correspond ..."
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Abstract. Recently S. Merkulov [Mer04, Mer05, Mer06] established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as corresponding to representations of the cobar construction on the Koszul dual of a certain quadratic operad. In this paper we prove, using the PBWbasis method of E. Hoffbeck [Hof08], that the operad governing Nijenhuis structures is Koszul, thereby showing that Nijenhuis structures correspond to representations of the minimal resolution of this operad. We also construct an operad such that representations of its minimal resolution in a vector space V are in onetoone correspondence with pairs of compatible Nijenhuis structures on the formal manifold associated to V.
On PostLie Algebras, Lie–Butcher Series and Moving Frames
"... Abstract PreLie (or Vinberg) algebras arise from flat and torsionfree connections on differential manifolds. These algebras have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic ..."
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Abstract PreLie (or Vinberg) algebras arise from flat and torsionfree connections on differential manifolds. These algebras have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on preLie algebras, used in numerical analysis to study geometric properties of flows on Euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of postLie algebras, a generalization of preLie algebras. Whereas preLie algebras are intimately associated with Euclidean geometry, postLie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan’s method of moving frames. Lie–Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie–Butcher series are founded on postLie algebras. The functorial relations between postLie algebras and their enveloping algebras, called Dalgebras, are explored. Furthermore, we develop new formulas for computations in free postLie
Theorem (Gerstenhaber).
"... For f ∈ Hom(V ⊗n, V) and g ∈ Hom(V ⊗m, V), binary product f ⋆ g:= n∑ i=1 ..."
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S II
"... Abstract. This paper is the followup of [MV08]. ..."
unknown title
, 2005
"... Bar constructions for topological operads and the Goodwillie derivatives of the identity ..."
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Bar constructions for topological operads and the Goodwillie derivatives of the identity