Rota-Baxter operators on unital algebras

TL;DR

This paper explores Rota-Baxter operators on various algebraic structures, establishing their properties, classifications, and introducing a new invariant called the rb-index, with specific results on matrix and Grassmann algebras.

Contribution

It provides a comprehensive analysis of Rota-Baxter operators on unital algebras, including classifications, correspondences with Yang-Baxter solutions, and introduces the rb-index as a new algebraic invariant.

Findings

All nonzero weight Rota-Baxter operators on Grassmann algebras are projections.

A one-to-one correspondence exists between solutions to the associative Yang-Baxter equation and weight-zero Rota-Baxter operators on matrix algebras.

The rb-index of the matrix algebra $M_n(F)$ is $2n-1$ in characteristic zero.

Abstract

We state that all Rota-Baxter operators of nonzero weight on 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. For an algebra $A$, we introduce its new invariant the rb-index $\mathrm{rb}(A)$ as the nilpotency index for 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.