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Baxter algebras originated in the following problem in fluctuation theory: Find the distribution functions of the maxima <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b1300801.png" /> of the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b1300802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b1300803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b1300804.png" /> of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b1300805.png" /> of independent identically-distributed random variables (cf. also [[Random variable|Random variable]]). A central result in this area is the Spitzer identity
+
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|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 function]]s 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 [[#References|[a2]]], [[#References|[a3]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b1300806.png" /></td> </tr></table>
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b1300807.png" /> is the characteristic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b1300808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b1300809.png" /> is the characteristic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008010.png" />. Spitzer's identity bears an uncanny resemblance to the Waring identity
+
where $q$ is a constant in $k$. A '''Baxter algebra''' is an algebra with a Baxter operator.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008011.png" /></td> </tr></table>
+
An example is the algebra of real-valued continuous functions on the interval $[0,1]$ with the integration operator
 +
$$
 +
PF(x) = \int_0^x f(t) dt \ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008012.png" /></td> </tr></table>
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The formula for [[integration by parts]] is identity (a1) with $q=0$.
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008013.png" /> are elementary symmetric functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008014.png" /> are power sum symmetric functions. The algebraic structure underlying both identities is a Baxter algebra. These algebras were defined by G.-C. Rota in [[#References|[a2]]], [[#References|[a3]]].
 
 
 
A Baxter operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008015.png" /> on an [[Algebra|algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008016.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008017.png" /> is a [[Linear operator|linear operator]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008018.png" /> to itself satisfying the identity
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008020.png" /> is a constant in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008021.png" />. A Baxter algebra is an algebra with a Baxter operator.
 
 
 
An example is the algebra of real-valued continuous functions on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008022.png" /> with the integration operator
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008023.png" /></td> </tr></table>
 
 
 
The formula for [[Integration by parts|integration by parts]] is identity (a1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008024.png" />.
 
 
 
Another example is the [[Banach algebra|Banach algebra]] of characteristic functions of distribution functions of random variables (cf. also [[Characteristic function|Characteristic function]]; [[Random variable|Random variable]]) with the Baxter operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008025.png" /> which sends the characteristic function of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008026.png" /> to the characteristic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008027.png" />. That is, if
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008028.png" /></td> </tr></table>
 
  
 +
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
 
then
 +
$$\label{eq:a2}
 +
P\xi(t) =  \int_0^\infty \exp(itx) dF(x) \ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
Given any [[endomorphism]] $E$ (that is, a linear operator satisfying $E(xy) = E(x)E(y)$) on an algebra $A$, the operator
 
+
$$
Given any [[Endomorphism|endomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008030.png" /> (that is, a linear operator satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008031.png" />) on an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008032.png" />, the operator
+
P = E + E^2 + \cdots = E(I-E)^{-1}
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008033.png" /></td> </tr></table>
+
is a Baxter operator if the infinite series converges. In particular, the $q$-integral
 
+
$$
is a Baxter operator if the infinite series converges. In particular, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008035.png" />-integral
+
Pf(t) = f(qt) + f(q^2t) + f(q^3t) + \cdots
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008036.png" /></td> </tr></table>
 
 
 
 
is a Baxter operator.
 
is a Baxter operator.
  
The standard Baxter algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008037.png" /> with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008038.png" /> is defined in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008041.png" /> be sequences such that the terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008042.png" /> are algebraically independent. On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008043.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008044.png" /> with coordinate-wise addition and multiplication generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008045.png" />, define the Baxter operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008046.png" /> by
+
The ''standard Baxter algebra'' over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008037.png" /> with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008038.png" /> is defined in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008041.png" /> be sequences such that the terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008042.png" /> are algebraically independent. On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008043.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008044.png" /> with coordinate-wise addition and multiplication generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008045.png" />, define the Baxter operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008046.png" /> by
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008047.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008047.png" /></td> </tr></table>
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Baxter,  "An analytic problem whose solution follows from a simple algebraic identity"  ''Pacific J. Math.'' , '''10'''  (1960)  pp. 731–742</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.-C. Rota,  "Baxter algebras and combinatorial identities I–II"  ''Bull. Amer. Math. Soc.'' , '''75'''  (1969)  pp. 325–334</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.-C. Rota,  "Baxter algebras: an introduction"  J.P.S. Kung (ed.) , ''Gian-Carlo Rota on Combinatorics'' , Birkhäuser  (1995)  pp. 504–512</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Baxter,  "An analytic problem whose solution follows from a simple algebraic identity"  ''Pacific J. Math.'' , '''10'''  (1960)  pp. 731–742</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  G.-C. Rota,  "Baxter algebras and combinatorial identities I–II"  ''Bull. Amer. Math. Soc.'' , '''75'''  (1969)  pp. 325–334</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  G.-C. Rota,  "Baxter algebras: an introduction"  J.P.S. Kung (ed.) , ''Gian-Carlo Rota on Combinatorics'' , Birkhäuser  (1995)  pp. 504–512</TD></TR>
 +
</table>
 +
 
 +
[[Category:TeX partially done]]

Revision as of 11:16, 28 November 2015

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 with generators is defined in the following way. Let , , be sequences such that the terms are algebraically independent. On the -algebra with coordinate-wise addition and multiplication generated by , define the Baxter operator by

The standard Baxter algebra is the smallest subalgebra of containing and closed under . Rota [a2], [a3] proved that the standard Baxter algebra is free in the category of Baxter algebras (cf. also Free algebra).

If is the sequence , then the st term in is the power sum symmetric function and the th term in , where there are occurrences of , is . Hence, the free Baxter algebra on one generator 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 is isomorphic to the algebra of polynomials in the variables . 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

When is the Baxter operator given in (a2), this identity is Spitzer's identity. When is the -integral, this identity becomes the Eulerian identity

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
How to Cite This Entry:
Baxter algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baxter_algebra&oldid=36835
This article was adapted from an original article by Joseph P.S. Kung (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article