Rota–Baxter algebra

In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map ${\displaystyle R}$ which satisfies the 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]

In the 1980s, the Rota-Baxter operator of weight 0 in the context of Lie algebras was rediscovered as the operator form of the classical Yang–Baxter equation,^[9] named after the well-known physicists Chen-Ning Yang and Rodney Baxter.

The study of Rota–Baxter algebras experienced a renaissance this century, beginning with several developments, in the algebraic approach to renormalization of perturbative quantum field theory,^[10] dendriform algebras, associative analogue of the classical Yang–Baxter equation^[11] and mixable shuffle product constructions.^[12]

Definition and first properties

Let ${\displaystyle k}$ be a commutative ring and let ${\displaystyle \lambda }$ be given. A linear operator ${\displaystyle R}$ on a ${\displaystyle k}$-algebra ${\displaystyle A}$ is called a Rota–Baxter operator of weight ${\displaystyle \lambda }$ if it satisfies the Rota–Baxter relation of weight ${\displaystyle \lambda }$:

${\displaystyle R(x)R(y)=R(R(x)y)+R(xR(y))+\lambda R(xy)}$

for all ${\displaystyle x,y\in A}$. Then the pair ${\displaystyle (A,R)}$ or simply ${\displaystyle A}$ is called a Rota–Baxter algebra of weight ${\displaystyle \lambda }$. In some literature, ${\displaystyle \theta =-\lambda }$ is used in which case the above equation becomes

${\displaystyle R(x)R(y)+\theta R(xy)=R(R(x)y)+R(xR(y)),}$

called the Rota-Baxter equation of weight ${\displaystyle \theta }$. The terms Baxter operator algebra and Baxter algebra are also used.

Let ${\displaystyle R}$ be a Rota–Baxter of weight ${\displaystyle \lambda }$. Then ${\displaystyle -\lambda Id-R}$ is also a Rota–Baxter operator of weight ${\displaystyle \lambda }$. Further, for ${\displaystyle \mu }$ in ${\displaystyle k}$, ${\displaystyle \mu R}$ is a Rota-Baxter operator of weight ${\displaystyle \mu \lambda }$.

Examples

Integration by parts

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

${\displaystyle I(f)(x)=\int _{0}^{x}f(t)dt\;.}$

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

${\displaystyle F(x)G(x)=\int _{0}^{x}f(t)G(t)dt+\int _{0}^{x}F(t)g(t)dt\;.}$

In other words

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

which shows that ${\displaystyle I}$ 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.

Bohnenblust–Spitzer identity