Free pre-Lie family algebras

Manchon, Dominique; Zhang, Yuanyuan

doi:10.48550/arxiv.2003.00917

Cited by 6 publications

(14 citation statements)

References 25 publications

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“…We shall say that pV, ˝q is a Φ-prelie algebra if, for any x, y, z P V , for any a, b P A, using Sweedler's notation (17) for Φ:…”

Section: Definitionmentioning

confidence: 99%

“…• If pΩ, ‹q is a commutative semigroup, taking EASpΩ, ‹q (which is a CEDS), we obtain Ω-family prelie algebras of [17].…”

Section: For Examplementioning

confidence: 99%

“…Another way is the use of one or more semigroup structures on Ω: this it the family parametrization. For example, family Rota-Baxter algebras, dendriform, pre-Lie algebras are introduced and studied in [20,21,17]. A way to obtain both these parametrizations for dendriform algebras is introduced in [10], with the help of a generalization of diassociative semigroups, namely extended diassociative semigroups (EDS), and a two-parameters version for dendriform algebras and pre-Lie algebras is described in [13].…”

Section: Introductionmentioning

confidence: 99%

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Generalized prelie and permutative algebras

Foissy

2021

Preprint

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We study generalizations of pre-Lie algebras, where the free objects are based on rooted trees which edges are typed, instead of usual rooted trees, and with generalized pre-Lie products formed by graftings. Working with a discrete set of types, we show how to obtain such objects when this set is given an associative commutative product and a second product making it a commutative extended semigroup. Working with a vector space of types, these two products are replaced by a bilinear map Φ which satisfies a braid equation and a commutation relation. Examples of such structures are defined on sets, semigroups, or groups.These constructions define a family of operads PreLie Φ which generalize the operad of pre-Lie algebras PreLie. For any embedding from PreLie into PreLie φ , we construct a family of pairs of cointeracting bialgebras, based on typed and decorated trees: the first coproduct is given by an extraction and contraction process, the types being modified by the action of Φ; the second coproduct is given by admissible cuts, in the Connes and Kreimer's way, with again types modified by the action of Φ.We also study the Koszul dual of PreLie Φ , which gives generalizations of permutative algebras.

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“…We shall say that pV, ˝q is a Φ-prelie algebra if, for any x, y, z P V , for any a, b P A, using Sweedler's notation (17) for Φ:…”

Section: Definitionmentioning

confidence: 99%

“…• If pΩ, ‹q is a commutative semigroup, taking EASpΩ, ‹q (which is a CEDS), we obtain Ω-family prelie algebras of [17].…”

Section: For Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized prelie and permutative algebras

Foissy

2021

Preprint

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“…Another way is to use one or more semigroup structures on Ω: this it the family parametrization. In this spirit, family Rota-Baxter algebras, dendriform, prelie algebras are introduced and studied in [20,21,12]. A way to obtain both these parametrizations for dendriform algebras is introduced in [6], with the help of a generalization of diassociative semigroups, called extended diassociative semigroups (briefly, EDS).…”

Section: Introductionmentioning

confidence: 99%

Generalized associative algebras

Foissy¹

2021

Preprint

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We study diverse parametrized versions of the operad of associative algebra, where the parameter are taken in an associative semigroup Ω (generalization of matching or family associative algebras) or in its cartesian square (two-parameters associative algebras). We give a description of the free algebras on these operads, study their formal series and prove that they are Koszul when the set of parameters is finite. We also study operadic morphisms between the operad of classical associative algebras and these objects.

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“…Another way is the use of one or more semigroup structures on Ω: this it the family parametrization. For example, family Rota-Baxter algebras, dendriform, pre-Lie algebras are introduced and studied in [11,12,7]. A way to obtain both these parametrizations for dendriform algebras is introduced in [3], with the help of a generalization of diassociative semigroups, namely extended diassociative semigroups (EDS).…”

Section: Introductionmentioning

confidence: 99%

On extended associative semigroups

Foissy

2021

Preprint

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We study extended associative semigroups (briefly, EAS), an algebraic structure used to define generalizations of the operad of associative algebras, and the subclass of commutative extended diassociative semigroups (briefly, CEDS), which are used to define generalizations of the operad of pre-Lie algebras. We give families of examples based on semigroups or on groups, as well as a classification of EAS of cardinality two. We then define linear extended associative semigroups as linear maps satisfying a variation of the braid equation. We explore links between linear EAS and bialgebras and Hopf algebras. We also study the structure of nondegenerate finite CEDS and show that they are obtained by semidirect and direct products involving two groups.

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Free pre-Lie family algebras

Cited by 6 publications

References 25 publications

Generalized prelie and permutative algebras

Generalized prelie and permutative algebras

Generalized associative algebras

On extended associative semigroups

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