Report title: Reynolds algebras and their free objects from bracketed words and rooted trees

Reporter: Happy Professor Lanzhou University

Reporting time: 10:30-11:30, December 10, 2020

Report location: Tencent Conference: 892522500

School contact: Tang Rong tangrong@jlu.edu.cn

Report summary:

The study of Reynolds algebras has its origin in the well-known work of O. Reynolds on fluid dynamics in 1895 and has since found broad applications. It also has close relationship with important linear operators such as algebra endomorphisms, derivations and Rota-Baxter operators . Many years ago G. Birkhoff suggested an algebraic study of Reynolds operators, including the corresponding free algebras. We carry out such a study in this talk. We first provide examples and properties of Reynolds operators, including a multi-variant generalization of the Reynolds identity. We then construct the free Reynolds algebra on a set. For this purpose, we identify a set of bracketed words called Reynolds words which serves as the linear basis of the free Reynolds algebra. A combinatorial interpretation of Reynolds words is given in terms of rooted trees without super crowns. The closure of the Reynolds words under concatenation gives the algebra structure on the space spanned by Reynolds words. The n a linear operator is defined on this algebra such that the Reynolds identity and the desired universal property are satisfied.

Brief introduction of the speaker:

Glad, PhD, professor and doctoral supervisor of Lanzhou University. In July 2010, he obtained his Ph.D degree from the School of Mathematics and Statistics of Lanzhou University and has stayed at the school until now. From August 2015 to August 2016, I visited Rutgers University in the United States, under the tutelage of Professor Guo Li, an international leader in Rota-Baxter algebra. Mainly engaged in research in the fields of Rota-Baxter algebra and algebraic combination, and published more than 40 SCI academic papers in International Journals such as Journal of Algebra, Journal of Pure and Applied Algebra. Won the second prize of the Gansu Provincial Natural Science Award, presided over the Tianyuan Foundation of Mathematics, Youth Science Foundation, National Natural Science Foundation of China and Gansu Natural Science Foundation of China, participated in 2 National Natural Science Foundation of China and Gansu Natural Science Foundation of China. 1 item, one monograph published.