Free pre-Lie family algebras. (English) Zbl 07835499
Summary: 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 Chapoton and Livernet. In the end, we construct Zinbiel and pre-Poisson family algebras and generalize results of Aguiar.
MSC:
| 16W99 | Associative rings and algebras with additional structure |
| 16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |
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| [39] | Communicated by Adrian Tanasȃ Received 18 April 2021. Yuanyuan Zhang School of Mathematics and Statistics, Henan University, 475004 Kaifeng, P. R. China; zhangyy17@henu.edu.cn Dominique Manchon Laboratoire de Mathématiques Blaise Pascal, CNRS-Université Clermont-Auvergne, 3 place Vasarély, CS 60026, 63178 Aubière, France; dominique.manchon@uca.fr |
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