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A remark on the Morita theorem for operads

Alexandru E. Stanculescu (2011)

Archivum Mathematicum

We extend a result of M. M. Kapranov and Y. Manin concerning the Morita theory for linear operads. We also give a cyclic operad version of their result.

Algèbres enveloppantes à homotopie près, homologies et cohomologies

Ridha Chatbouri (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

On présente une définition et une construction unifée des homologies et cohomologies d’algèbres et de modules sur ces algèbres et de modules sur ces algèbres dans le cas d’algèbres associatives ou commutatives ou de Lie ou de Gertsenhaber. On sépare la construction linéaire des cogèbres ou bicogèbres qui traduisent les symétries des relations de définition de la structure de la partie structure qui apparaît ici comme une codérivation de degré 1 et de carré nul de la cogèbre ou de la bicogèbre.

Categorification of Hopf algebras of rooted trees

Joachim Kock (2013)

Open Mathematics

We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ℕ) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ℤ and collapsing H 0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring...

Cohomology operations and the Deligne conjecture

Martin Markl (2007)

Czechoslovak Mathematical Journal

The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.

De Rham cohomology and homotopy Frobenius manifolds

Vladimir Dotsenko, Sergey Shadrin, Bruno Vallette (2015)

Journal of the European Mathematical Society

We endow the de Rham cohomology of any Poisson or Jacobi manifold with a natural homotopy Frobenius manifold structure. This result relies on a minimal model theorem for multicomplexes and a new kind of a Hodge degeneration condition.

Embedding of dendriform algebras into Rota-Baxter algebras

Vsevolod Gubarev, Pavel Kolesnikov (2013)

Open Mathematics

Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform...

From left modules to algebras over an operad: application to combinatorial Hopf algebras

Muriel Livernet (2010)

Annales mathématiques Blaise Pascal

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from 𝕊 -modules to graded vector spaces in the context of algebras over an operad and derive the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for those Hopf algebras.Let 𝒪 denote the forgetful functor from 𝕊 -modules to graded vector spaces. Left modules over an operad 𝒫 are treated as 𝒫 -algebras in the category of 𝕊 -modules. We generalize the results obtained...

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