The Pre-Lie Operad as a Deformation of Nap

Saidi, Anouar

doi:10.1142/s021949881350076x

Cited by 11 publications

(10 citation statements)

References 18 publications

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“…We shall now see that the same is true for the operad PreLie C , so that the operad structure of PreLie C "deforms" the operad structure of NAP C by adding lower terms. We believe that the latter observation can be made in a precise statement using a formalism similar to that of [42,43]. Proposition 3.…”

Section: Remarkmentioning

confidence: 72%

Enriched pre-Lie operads and freeness theorems

Dotsenko¹,

Foissy²

2020

Preprint

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In this paper, we study the operad of C-enriched pre-Lie algebras, defined for any Hopf cooperad C; it slightly generalises a similar notion defined by Calaque and Willwacher to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding operads of enriched pre-Lie algebras; we prove a criterion for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré-Birkhoff-Witt type theorem for universal enveloping brace algebras of pre-Lie algebras.

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Section: Remarkmentioning

confidence: 72%

Enriched pre-Lie operads and freeness theorems

Dotsenko¹,

Foissy²

2020

Preprint

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“…Unlike PLie, NAP is not free as a nonsymmetric operad (it is easy to find a nontrivial relation in degree 2 for instance). Some links between PLie and NAP have been exploited in [Saï14]. The operad of noncrossing trees NCT was defined in [Cha07] as a suboperad of a bigger operad Mould, the operad of mould.…”

Section: About the Presented Examples Of Operadsmentioning

confidence: 99%

Nonsymmetric operads in combinatorics

Giraudo

2021

Preprint

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A. Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. The theory of operads, together with the algebraic setting and the tools accompanying it, promises advances in these two areas. On the one hand, operads provide a useful abstraction of formal expressions, and also, provide connections with the theory of rewrite systems. On the other hand, a lot of operads involving combinatorial objects highlight some of their properties and allow to discover new ones. This book presents the theory of nonsymmetric operads under a combinatorial point of view. It portrays the main elements of this theory and the links it maintains with several areas of computer science and combinatorics. A lot of examples of operads appearing in combinatorics are studied and some constructions relating operads with known algebraic structures are presented. The modern treatment of operads consisting in considering the space of formal power series associated with an operad is developed. Enrichments of nonsymmetric operads as colored, cyclic, and symmetric operads are reviewed. This text is addressed to any computer scientist or combinatorist who looks a complete and a modern description of the theory of nonsymmetric operads. Evenly, this book is intended to an audience of algebraists who are looking for an original point of view fitting in the context of combinatorics.April 27, 2021 3.2. Pros Main pros Bibliography(2) Chapter 2 is devoted to general treelike structures. It presents syntax trees, that are sorts of trees appearing in the study of nonsymmetric operads. Rewrite systems on syntax trees are considered and tools to prove their termination or confluence are provided.(3) Chapter 3 concerns algebraic structures defined on the linear span of collections. These vector spaces are called polynomial spaces. It presents also the notion of biproducts on polynomial spaces and of types of bialgebras. Classical types of bialgebras appearing in combinatorics are given: associative, dendriform, and pre-Lie algebras, and Hopf bialgebras.(4) Chapter 4 presents nonsymmetric operads and related notions. It exposes the notions of algebras over operads, free operads, presentations by generators and relations, Koszul duality, and Koszulity. Several examples of operads appearing in algebraic combinatorics are reviewed.(5) Chapter 5 contains generalizations, applications, and apertures of the theory of nonsymmetric operads. It reviews three topics in this vein. First, formal power series on nonsymmetric operads are considered and applications to enumeration are provided. Next, enrichments of nonsymmetric operads are discussed: colored operads, cyclic operads, and symmetric operads. Finally, it provides an overview of product categories, a generalization of operads wherein elements can ha...

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“…Proof. We adapt the proof from [31,Theorem 10], where the associativity is proved in the slightly different context of a "current-preserving" version of the pre-Lie operad. Let T ∈ T Ω (A), U ∈ T Ω (B) and W ∈ T Ω (C) where A, B, C are three finite sets.…”

Section: 2mentioning

confidence: 99%

Free pre-Lie family algebras

Manchon,

Zhang

2020

Preprint

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In this paper, we first define the pre-Lie family algebra associated to a dendriform family algebra in the case of a commutative semigroup. Then we construct a pre-Lie family algebra via typed decorated rooted trees, and we prove the freeness of this pre-Lie family algebra. We also construct pre-Lie family operad in terms of typed labeled rooted trees, and we obtain that the operad of pre-Lie family algebras is isomorphic to the operad of typed labeled rooted trees, which generalizes the result of F. Chapoton and M. Livernet. In the end, we construct Zinbiel and pre-Poisson family algebras and generalize results of M. Aguiar.

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The Pre-Lie Operad as a Deformation of Nap

Abstract: We will define a family of multigraded operads O λ dependent on a scalar parameter λ which is a deformation of the operad O 0 . Forgetting the multi-graduation gives back the pre-Lie operad for λ = 1 and the NAP operad for λ = 0.

Cited by 11 publications

References 18 publications

Enriched pre-Lie operads and freeness theorems

Enriched pre-Lie operads and freeness theorems

Nonsymmetric operads in combinatorics

Free pre-Lie family algebras

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