Baxter algebra

Baxter algebras originated in the following problem in fluctuation theory: Find the distribution functions of the maxima $\max\{0, S_1, \ldots, S_n \}$ of the partial sums $S_0 = 0$, $S_1=X_1$, $S_2=X_1 + X+2$, $\ldots$, $S_n = X_1+\cdots+X_n$, of a sequence $X_i$ of independent identically-distributed random variables (cf. also Random variable). A central result in this area is the Spitzer identity $$ \sum_{n=0}^\infty \phi_n(t) \lambda^n = \exp\left[{ \psi_k(t) \frac{\lambda^k}{k} }\right]\ , $$ where $\phi_n(t)$ is the characteristic function of $\max\{0, S_1, \ldots, S_n \}$ and $\psi_k(t)$ is the characteristic function of $\max\{0, S_k \}$. Spitzer's identity bears an uncanny resemblance to the Waring identity $$ \sum_{n=0}^\infty e_n(x_1,x_2,\ldots) \lambda^n = \exp\left[{ -\sum_{k=1}^\infty (-1)^k p_k(x_1,x_2,\ldots) \frac{\lambda^k}{k} }\right] $$ where $e_n(x_1,x_2,\ldots)$ are elementary symmetric functions and $p_k(x_1,x_2,\ldots)$ are power sum symmetric functions. The algebraic structure underlying both identities is a Baxter algebra. These algebras were defined by G.-C. Rota in [a2], [a3].

A Baxter operator $P$ on an algebra $A$ over a field $k$ is a linear operator from $A$ to itself satisfying the identity $$\label{eq:a1} P(xPy) + P(yPx) = (Px)(Py) + q P(xy) $$

where $q$ is a constant in $k$. A Baxter algebra is an algebra with a Baxter operator.

An example is the algebra of real-valued continuous functions on the interval $[0,1]$ with the integration operator $$ PF(x) = \int_0^x f(t) dt \ . $$

The formula for integration by parts is identity (a1) with $q=0$.

Another example is the Banach algebra of characteristic functions of distribution functions of random variables (cf. also Characteristic function; Random variable) with the Baxter operator $P$ which sends the characteristic function of a random variable $X$ to the characteristic function of $\max\{0,X\}$. That is, if $$ \xi(t) = \int_{-\infty}^\infty \exp(itx) dF(x) $$ then $$\label{eq:a2} P\xi(t) = \int_0^\infty \exp(itx) dF(x) \ . $$

Given any endomorphism $E$ (that is, a linear operator satisfying $E(xy) = E(x)E(y)$) on an algebra $A$, the operator $$ P = E + E^2 + \cdots = E(I-E)^{-1} $$ is a Baxter operator if the infinite series converges. In particular, the $q$-integral $$ Pf(t) = f(qt) + f(q^2t) + f(q^3t) + \cdots $$ is a Baxter operator.

The standard Baxter algebra over a field $F$ with generators $x,y,\ldots$ is defined in the following way. Let $x = (x_1,x_2,\ldots)$, $y = (y_1,y_2,\ldots)$, $\ldots$ be sequences such that the terms $x_1,x_2,\ldots,y_1,y_2,\ldots$ are algebraically independent. On the $F$-algebra $A$ with coordinate-wise addition and multiplication generated by $x,y,\ldots$, define the Baxter operator $P$ by $$ P(u_1,u_2,u_3,\ldots) = (0,u_1,u_1+u_2,u_1+u_2+u_3,\ldots) $$

The standard Baxter algebra $B$ is the smallest subalgebra of $A$ containing $x,y,\ldots$ and closed under $P$. Rota [a2], [a3] proved that the standard Baxter algebra is free in the category of Baxter algebras (cf. also Free algebra).

If $x$ is the sequence $(x_1,x_2,\ldots)$, then the $(k+1)$-st term in $P(x^n)$ is the power sum symmetric function $x_1^n+\cdots+x_k^n$ and the $k$-th term in $P(xP(\ldots(xPx)\ldots))$, where there are $n$ occurrences of $P$, is $e_n(x_1,\ldots,x_k)$. Hence, the free Baxter algebra on one generator $x$ is isomorphic to the algebra of symmetric functions (cf. also Symmetric function). Because the elementary symmetric functions are algebraically independent, the free Baxter algebra in one generator $x$ is isomorphic to the algebra of polynomials in the variables $x,Px,P(xPx),\ldots$. This solves the word problem (cf. also Identity problem) for Baxter algebras with one generator. The word problem for Baxter algebras with more than one generator is solved in a similar way by P. Cartier. In particular, an identity amongst symmetric functions can be translated into an identity satisfied by all Baxter algebras on one generator. For example, writing Waring's identity in terms of Baxter operators, one obtains $$ \sum_{n=0}^\infty P(xP(\ldots(xPx)\ldots)) \lambda^n = \exp\left[{ -P\sum_{k=1}^\infty (-1)^k t^k \frac{\lambda^k}{k} }\right] = \exp[P \log(1+tx)] \ . $$

When $P$ is the Baxter operator given in (a2), this identity is Spitzer's identity. When $P$ is the $q$-integral, this identity becomes the Eulerian identity $$ \sum_{n=1}^\infty \frac{ t^n q^{n(n+1)/2} }{ (1-q)\cdots(1-q^n) } = \prod_{k=1}^\infty (1+q^k t) \ . $$

References

[a1]	G. Baxter, "An analytic problem whose solution follows from a simple algebraic identity" Pacific J. Math. , 10 (1960) pp. 731–742
[a2]	G.-C. Rota, "Baxter algebras and combinatorial identities I–II" Bull. Amer. Math. Soc. , 75 (1969) pp. 325–334
[a3]	G.-C. Rota, "Baxter algebras: an introduction" J.P.S. Kung (ed.) , Gian-Carlo Rota on Combinatorics , Birkhäuser (1995) pp. 504–512