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Zhang, Yuanyuan; Gao, Xing; Manchon, Dominique

Free Rota-Baxter family algebras and free (tri)dendriform family algebras. (English) Zbl 1541.16038

Algebr. Represent. Theory 26, No. 6, 2967-3002 (2023).
The notion of a Rota-Baxter family, which is proposed by L. Guo [J. Algebr. Comb. 29, No. 1, 35–62 (2009; Zbl 1227.05271)], appears in the algebraic aspects of renormalization in quantum field theory. This paper gives another construction of free Rota-Baxter family algebras by the planar rooted trees (not planar rooted forests), which are both angularly decorated by a set of generators and typed by a semigroup. As an application, a new construction of the free Rota-Baxter algebra is provided in terms of angularly decorated planar rooted trees (not forests), different from the known construction via angularly decorated planar rooted forests. Furthermore, the authors prove that the free dendriform (resp. tridendriform) family algebra can be embedded into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, this paper shows that the universal enveloping algebra of the free (tri)dendriform family algebra is just the free Rota-Baxter family algebra.
Reviewer: Shanghua Zheng (Nanchang)

Cited in 2 Documents

MSC:

16W99 Associative rings and algebras with additional structure
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
08B20 Free algebras
16T30 Connections of Hopf algebras with combinatorics

Keywords:

Rota-Baxter family algebra; (tri)dendriform family algebra; typed decorated Planar rooted trees; free Rota-Baxter family algebras; free (tri)dendriform family algebras

Citations:

Zbl 1227.05271
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Full Text: DOI arXiv

References:

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