Generalized prelie and permutative algebras
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866908603780169728 |
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| author | Foissy, Loïc |
| author_facet | Foissy, Loïc |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_00909 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Generalized prelie and permutative algebras Foissy, Loïc Rings and Algebras 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. |
| title | Generalized prelie and permutative algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2104.00909 |