arXiv subjects

Jean-Louis Loday

Publications and source records attributed to Jean-Louis Loday.

At least 19 recordsLinked to original sources

arXiv2012

Permutads

We unravel the algebraic structure which controls the various ways of computing the word ((xy)(zt)) and its siblings. We show that it gives rise to a new type of operads, that we call permutads. It turns out that this notion is equivalent to the notion of "shuffle algebra" introduced by the second author. It is also very close to the notion of "shuffle operad" introduced by V. Dotsenko and A. Khoroshkin. It can be seen as a noncommutative version of the notion of nonsymmetric operads. We show that the role of the associahedron in the theory of operads is played by the permutohedron in the theory of permutads.

math.QA↗

arXiv2012

Operadic construction of the renormalization group

First, we give a functorial construction of a group associated to a symmetric operad. Applied to the endomorphism operad it gives the group of formal diffeomorphisms. Second, we associate a symmetric operad to any family of decorated graphs stable by contraction. In the case of Quantum Field Theory models it gives the renormalization group. As an example we get an operadic interpretation of the group of "diffeographisms" attached to the Connes-Kreimer Hopf algebra.

math-ph↗

arXiv2012

Exponential series without denominators

For a commutative algebra which comes from a Zinbiel algebra the exponential series can be written without denominators. When lifted to dendriform algebras this new series satisfies a functional equation analogous to the Baker-Campbell-Hausdorff formula. We make it explicit by showing that the obstruction series is the sum of the brace products. In the multilinear case we show that the role the Eulerian idempotent is played by the iterated pre-Lie product.

math.RA↗

arXiv2011

Free loop space and homology

The aim of this paper is to explain the relationship between the (co)homology of the free loop space and the Hochschild homology of its singular cochain algebra. We introduce all the relevant technical tools, namely simplicial and cyclic objects, and we provide the various steps of the proofs, which are scattered around in the literature. This paper can be seen as a companion to the paper "Cyclic homology and equivariant homology" by J.D.S. Jones dealing with the cyclic case.

math.AT↗

arXiv2011

Some problems in operad theory

This is a list of some problems and conjectures related to various types of algebras, that is to algebraic operads. Some comments and hints are included.

math.RA↗

arXiv2011

Dichotomy of the addition of natural numbers

This is an elementary presentation of the arithmetic of trees. We show how it is related to the Tamari poset. In the last part we investigate various ways of realizing this poset as a polytope (associahedron), including one inferred from Tamari's thesis.

math.RA↗

arXiv2011

The YY game

We introduce a new one-person game similar to the Sudoku game. It is based on combinatorial objects called planar binary rooted trees. It is related to the four color conjecture. Its mathematical analysis makes use of the Tamari poset, hence the Stasheff associahedron.

math.CO↗

arXiv2010

The symmetric operation in a free pre-Lie algebra is magmatic

A pre-Lie product is a binary operation whose associator is symmetric in the last two variables. As a consequence its antisymmetrization is a Lie bracket. In this paper we study the symmetrization of the pre-Lie product. We show that it does not satisfy any other universal relation than commutativity. It means that the map from the free commutative-magmatic algebra to the free pre-Lie algebra induced by the symmetrization of the pre-Lie product is injective. This result is in contrast with the associative case, where the symmetrization gives rise to the notion of Jordan algebra. We first give a self-contained proof. Then we give a proof which uses the properties of dendriform and duplicial algebras.

math.RA↗

arXiv2010

On the operad of associative algebras with derivation

We study the operad of associative algebras equipped with a derivation. We show that it is determined by polynomials in several variables and substitution. Replacing polynomials by rational functions gives an operad which is isomorphic to the operad of "moulds". It provides an efficient environment for doing integro-differential calculus. Interesting variations are obtained by using formal group laws. The preceding case corresponds to the additive formal group law. We unravel the notion of homotopy associative algebra with derivation in the spirit of Kadeishvili's work.

math.RA↗

arXiv2009

Combinatorial Hopf algebras

We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). The choice of such an isomorphism implies the existence a finer algebraic structure on the Hopf algebra and on the indecomposables (resp. the primitives). For instance a cofree-cocommutative right-sided combinatorial Hopf algebra is completely determined by its primitive part which is a pre-Lie algebra. The key example is the Connes-Kreimer Hopf algebra. The study of all these combinatorial Hopf algebra types gives rise to several good triples of operads. It involves the operads: dendriform, pre-Lie, brace, and variations of them.

math.QA↗

arXiv2008

Generalized bialgebras and triples of operads

We introduce the notion of generalized bialgebra, which includes the classical notion of bialgebra (Hopf algebra) and many others. We prove that, under some mild conditions, a connected generalized bialgebra is completely determined by its primitive part. This structure theorem extends the classical Poincaré-Birkhoff-Witt theorem and the Cartier-Milnor-Moore theorem, valid for cocommutative bialgebras, to a large class of generalized bialgebras. Technically we work in the theory of operads which permits us to give a conceptual proof of our main theorem. It unifies several results, generalizing PBW and CMM, scattered in the literature. We treat many explicit examples and suggest a few conjectures.

math.QA↗

arXiv2008

Parastatistics Algebra, Young Tableaux and the Super Plactic Monoid

The parastatistics algebra is a superalgebra with (even) parafermi and (odd) parabose creation and annihilation operators. The states in the parastatistics Fock-like space are shown to be in one-to-one correspondence with the Super Semistandard Young Tableaux (SSYT) subject to further constraints. The deformation of the parastatistics algebra gives rise to a monoidal structure on the SSYT which is a super-counterpart of the plactic monoid.

math-ph↗

arXiv2007

Parastatistics Algebra and Super Semistandard Young Tableaux

We consider the parastatistics algebra with both parabosonic and parafermionic operators and show that the states in the universal parastatistics Fock space are in bijection with the Super Semistandard Young Tableaux (SSYT). Using deformation of the parastatistics algebra we get a monoid structure on SSYT which is a super version of the plactic monoid.

math-ph↗

arXiv2007

The diagonal of the Stasheff polytope

We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and the operad A-infinity is a retract by deformation of it. We compare these constructions with analogous constructions due to Saneblidze-Umble and Markl-Shnider based on the Boardman-Vogt cubical decomposition of the Stasheff polytope.

math.AT↗

arXiv2007

On the algebra of quasi-shuffles

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 quasi-shuffle 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 (CTD-algebras). This result is part of a structure theorem for CTD-bialgebras which are associative as coalgebras and whose primitive part is commutative. In other words, there is a good triple of operads $(As, CTD, Com)$ analogous to $(Com, As, Lie)$.

math.QA↗

arXiv2006

Coassociative magmatic bialgebras and the Fine numbers

We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n-2 operations of arity n. The dimension of the space of all the n-ary operations of this primitive operad turns out to be the Fine number F_{n-1}. In short, the triple of operads (As, Mag, MagFine) is good.

math.RA↗

arXiv2005

Parking functions and triangulation of the associahedron

We show that a minimal triangulation of the associahedron (Stasheff polytope) of dimension n is made of (n+1)^{n-1} simplices. We construct a natural bijection with the set of parking functions from a new interpretation of parking functions in terms of shuffles.

math.AT↗

arXiv2005

On restricted Leibniz algebras

In this paper we prove that in prime characteristic there is a functor $-_{p-Leib}$ from the category of diassociative algebras to the category of restricted Leibniz algebras, generalizing the functor from associative algebras to restricted Lie algebras. Moreover we define the notion of restricted universal enveloping diassociative algebra $Ud_{p}(g)$ of a restricted Leibniz algebra $g$ and we show that $Ud_{p}$ is left adjoint to the functor $-_{p-Leib}$. We also construct the restricted enveloping algebra, which classifies the restricted Leibniz modules. In the last section we put a restricted pre-Lie structure on the tensor product of a Leibniz algebra by a Zinbiel algebra.

math.RA↗