Spectrum of Rota-Baxter operators

2020/06/04 by Vsevolod Gubarev, Gubarev, Vsevolod
Computer Science · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · doi:10.48550/arxiv.2006.02654

openalex publication_date 2020/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the spectrum of every Rota-Baxter operator of weight λ on a unital algebraic (not necessarily associative) algebra over a field of characteristic zero is a subset of \0,-λ\. For a finite-dimensional unital algebra the same statement is shown to hold without a restriction on the characteristic of the ground field. Based on these results, we define the Rota-Baxter λ-index rb_λ(A) of an algebra A as the infimum of the degrees of minimal polynomials of all Rota-Baxter operators of weight λ on A. We calculate the Rota-Baxter λ-index for the matrix algebra M_n(F), char F = 0: it is shown that rb_λ(M_n(F)) = 2n-1.

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