A Page on Rota-Baxter Algebra

Summary: A Rota-Baxter algebra is an associative algebra with a linear operator that generalizes the algebra of continuous functions with the integral operator. More precisely, for a given commutative ring k and &lambda in k, a Rota-Baxter k-algebra (of weight &lambda) is a k-algebra R together with a k-linear operator P on R such that
P(x)P(y)=P(P(x)y) + P(xP(y)) + &lambdaP(xy)
(1)
for all x,y in R. Such a linear operator is called a Rota-Baxter operator (of weight &lambda). It is easy to see that the Rota-Baxter operator is an abstraction of the integration by parts formula. Note that the relation (1) still makes sense when the associative algebra R is replaced by a k-module with a bilinear binary operation, such as the Lie bracket. The study of Rota-Baxter algebra originated from the probability study of Glenn Baxter in 1960 and was developed further by Cartier and the school of Rota in the 1960s and 1970s. Independently, this structure appeared in the Lie algebra context as the operator form of the classical Yang-Baxter equation started in the 1980s. Since the late 1990s, Rota-Baxter algebra has experienced a quite remarkable renascence and found important theoretical developments and applications in mathematical physics, operads, number theory and combinatorics.
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