TY - JOUR

T1 - Spectrum of Rota–Baxter operators

AU - Gubarev, Vsevolod

PY - 2025

Y1 - 2025

N2 - Rota–Baxter operators have been studied since 1960, and there are lots of their applications in mathematical physics, number theory, and noncommutative geometry. We state a surprisingly general property of such operators: the spectrum of every Rota–Baxter operator of weight λ on a finite-dimensional unital (not necessarily associative) algebra is a subset of {0, −λ}. We even extend this result to the case of infinite-dimensional algebraic algebras (when char F = 0). Based on these results, we define a new invariant of an algebra: the Rota–Baxter λ-index rbλ(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 compute the Rota–Baxter λ-index for the matrix algebra Mn(F), char F = 0: it is shown that rbλ(Mn(F)) = 2n − 1.

AB - Rota–Baxter operators have been studied since 1960, and there are lots of their applications in mathematical physics, number theory, and noncommutative geometry. We state a surprisingly general property of such operators: the spectrum of every Rota–Baxter operator of weight λ on a finite-dimensional unital (not necessarily associative) algebra is a subset of {0, −λ}. We even extend this result to the case of infinite-dimensional algebraic algebras (when char F = 0). Based on these results, we define a new invariant of an algebra: the Rota–Baxter λ-index rbλ(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 compute the Rota–Baxter λ-index for the matrix algebra Mn(F), char F = 0: it is shown that rbλ(Mn(F)) = 2n − 1.

KW - Rota–Baxter module

KW - Rota–Baxter operator

KW - associative Yang–Baxter equation

KW - matrix algebra

UR - https://www.mendeley.com/catalogue/37217b16-5303-3fe6-b82f-d9a6739dfd2f/

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-105007533210&origin=inward&txGid=1a0f111815987820b9dc714b07357e43

U2 - 10.1142/S0218196725500195

DO - 10.1142/S0218196725500195

M3 - Article

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

SN - 0218-1967

ER -