# Rota–Baxter algebra

In mathematics, a Rota–Baxter algebra is an algebra, usually over a field k, together with a particular k-linear map R which satisfies the weight-θ Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter[1] in the realm of probability theory. Baxter's work was further explored from different angles by Gian-Carlo Rota,[2][3][4] Pierre Cartier,[5] and Frederic V. Atkinson,[6] among others. Baxter’s derivation of this identity that later bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory.[7][8]

## Definition and first properties

Let A be a k-algebra with a k-linear map R on A and let θ be a fixed parameter in k. We call A a Rota-Baxter k-algebra and R a Rota-Baxter operator of weight θ if the operator R satisfies the following Rota–Baxter relation of weight θ:

$R(x)R(y) + \theta R(xy) = R(R(x)y + xR(y)).\,$

The operator R:= θ id − R also satisfies the Rota–Baxter relation of weight θ.

## Examples

Integration by Parts

Integration by parts is an example of a Rota–Baxter algebra of weight 0. Let $C(R)$ be the algebra of continuous functions from the real line to the real line. Let :$f(x) \in C(R)$ be a continuous function. Define integration as the Rota–Baxter operator

$I(f)(x) = \int_0^x f(t) dt \;.$

Let G(x) = I(g)(x) and F(x) = I(f)(x). Then the formula for integration for parts can be written in terms of these variables as

$F(x)G(x) = \int_0^x f(t) G(t) dt + \int_0^x F(t)g(t) dt \;.$

In other words

$I(f)(x)I(g)(x) = I(fI(g)(t))(x) + I(I(f)(t)g)(x) \; ,$

which shows that I is a Rota–Baxter algebra of weight 0.

## Spitzer identity

The Spitzer identity appeared is named after the American mathematician Frank Spitzer. It is regarded as a remarkable stepping stone in the theory of sums of independent random variables in fluctuation theory of probability. It can naturally be understood in terms of Rota–Baxter operators.