Rota-Baxter operators on unital algebras

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.

v3: 43 p; Introduction was rewritten; Theorem 4.13 and Remark 5.9 are new v2: 37 p; Theorem 5.21 is new

References in corpus (1)

• Rota-Baxter Operators on Quadratic Algebras

Cited by in corpus (8)

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• Classification of pre-Jordan Algebras and Rota-Baxter Operators on Jordan Algebras in Low Dimensions
• Injective Rota-Baxter operators of weight zero on $F[x]$
• Monomial Rota-Baxter operators on free commutative non-unital algebra
• Rota-Baxter operators of nonzero weight on the matrix algebra of order three
• Double Lie algebras of a nonzero weight