Free Rota–Baxter family algebras and (tri)dendriform family algebras

Yuanyuan Zhang and Xing Gao

Vol. 301 (2019), No. 2, 741–766

DOI: 10.2140/pjm.2019.301.741

Abstract

We construct free commutative Rota–Baxter family algebras, and then free noncommutative Rota–Baxter family algebras via the method of Gröbner–Shirshov bases. We introduce the concept of dendriform (resp. tridendriform) family algebras, and prove that Rota–Baxter family algebras of weight zero (resp. $\lambda$) induce dendriform (resp. tridendriform) family algebras. We also construct free commutative dendriform (resp. tridendriform) family algebras.

Keywords

Rota–Baxter algebra, Rota–Baxter family algebra, dendriform family algebra, tridendriform family algebra, Gröbner–Shirshov basis

Mathematical Subject Classification 2010

Primary: 08B20, 13P10, 16S10, 16W99

Milestones

Received: 30 July 2018

Revised: 27 October 2018

Accepted: 29 October 2018

Published: 24 October 2019