Moscow Mathematical Journal

Volume 21, Issue 2, April–June 2021 pp. 325–364.

Rota–Baxter Operators on Unital Algebras

Authors: V. Gubarev (1)
Author institution:(1) University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
Sobolev Institute of Mathematics, Acad. Koptyug ave. 4, 630090 Novosibirsk, Russia

Summary:

We state that all Rota–Baxter operators of nonzero weight on the Grassmann algebra over a field of characteristic zero are projections on a subalgebra along another one. We show the one-to-one correspondence between the solutions of associative Yang–Baxter equation and Rota–Baxter operators of weight zero on the matrix algebra $M_n(F)$ (joint with P. Kolesnikov).

We prove that all Rota–Baxter operators of weight zero on a unital associative (alternative, Jordan) algebraic algebra over a field of characteristic zero are nilpotent. We introduce a new invariant for an algebra $A$ called the RB-index $\mathrm{rb}(A)$ as the minimal nilpotency index of Rota–Baxter operators of weight zero on $A$. We show that $\mathrm{rb}(M_n(F)) = 2n-1$ provided that characteristic of $F$ is zero.

2020 Math. Subj. Class. 16W99, 17C20

Keywords: Rota–Baxter operator, Yang–Baxter equation, matrix algebra, Grassmann algebra, Faulhaber polynomial.

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