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Rota-Baxter algebra

Rota–Baxter algebras

Definition

Given a commutative unital ring kk, an associative unital kk-algebra AA is a Rota–Baxter algebra of weight λ\lambda if it is equipped with a Rota–Baxter operator P:AAP\colon A \to A, which is a kk-linear endomorphism such that

P(x)P(y)=P(P(x)y)+P(xP(y))+λP(xy) P(x) P(y) = P(P(x) y) + P(x P(y)) + \lambda P(x y)

Historical sources and motivation

Rota–Baxter algebras came historically from several sources, including enumerative combinatorics problems (Gian-Carlo Rota), qq-calculus (Jackson integral) and integrable systems (Baxter). More recently the application to the combinatorics of renormalization and Birkhoff decomposition have been found, showing that the essence of renormalization is not that crucially existing only in the presence of complex analysis.

References

A large bibliography on Rota–Baxter algebras can be found at Li Guo’s Rota–Baxter algebra page.

A generalization of Rota-Baxter operators to algebraic operads is found in connection to certain splitting phenomenon for operads:

There is also a categorification of the concept of Rota–Baxter algebra, namely the Rota–Baxter category, cf.

Among motivatring results, there are also Spitzer identities from