Namespaces
Variants
Actions

Difference between revisions of "Baxter algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX partially done)
 
(9 intermediate revisions by 2 users not shown)
Line 1: Line 1:
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''
+
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 variable]]s. 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]\ ,
+
\sum_{n=0}^\infty \phi_n(t) \lambda^n = \exp\left[{ \sum_{k=1}^\infty \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''
+
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 remarkable 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]
 
\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]
Line 18: Line 18:
 
An example is the algebra of real-valued continuous functions on the interval $[0,1]$ with the integration 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 \ .
+
Pf(x) = \int_0^x f(t) dt \ .
 
$$
 
$$
  
Line 42: Line 42:
 
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 $F$ 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
 +
$$
 +
P(u_1,u_2,u_3,\ldots) = (0,u_1,u_1+u_2,u_1+u_2+u_3,\ldots)
 +
$$
  
<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>
+
The standard Baxter algebra $B$ is the smallest subalgebra of $A$ containing $x,y,\ldots$ and closed under $P$. Rota [[#References|[a2]]], [[#References|[a3]]] proved that the standard Baxter algebra is free in the category of Baxter algebras (cf. also [[Free algebra]]).
  
The standard Baxter algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008048.png" /> is the smallest subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008049.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008050.png" /> and closed under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008051.png" />. Rota [[#References|[a2]]], [[#References|[a3]]] proved that the standard Baxter algebra is free in the category of Baxter algebras (cf. also [[Free algebra|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|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
 +
$$
 +
\sum_{n=0}^\infty P(xP(\ldots(xPx)\ldots)) \lambda^n = \exp\left[{ -P\sum_{k=1}^\infty (-1)^k t^k \frac{\lambda^k}{k} }\right] = \exp[P \log(1+tx)] \ .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008052.png" /> is the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008053.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008054.png" />st term in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008055.png" /> is the power sum symmetric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008056.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008057.png" />th term in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008058.png" />, where there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008059.png" /> occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008060.png" />, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008061.png" />. Hence, the free Baxter algebra on one generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008062.png" /> is isomorphic to the algebra of symmetric functions (cf. also [[Symmetric function|Symmetric function]]). Because the elementary symmetric functions are algebraically independent, the free Baxter algebra in one generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008063.png" /> is isomorphic to the algebra of polynomials in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008064.png" />. This solves the word problem (cf. also [[Identity problem|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 $P$ 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
 +
$$
 +
\sum_{n=1}^\infty \frac{ t^n q^{n(n+1)/2} }{ (1-q)\cdots(1-q^n) } = \prod_{k=1}^\infty (1+q^k t) \ .
 +
$$
  
<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/b13008065.png" /></td> </tr></table>
+
====References====
 
+
<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/b13008066.png" /></td> </tr></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 {{DOI|10.2140/pjm.1960.10.731}} {{ZBL|0095.12705}}</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 {{DOI|10.1090/S0002-9904-1969-12158-0}} {{ZBL|0319.05008}}</TD></TR>
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008067.png" /> is the Baxter operator given in (a2), this identity is Spitzer's identity. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008068.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130080/b13008069.png" />-integral, this identity becomes the Eulerian identity
+
<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: Introductory papers and commentaries'' , Birkhäuser  (1995) {{ISBN|3-7643-3713-3}} pp. 504–512 {{ZBL|0841.01031}}</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  Frank Spitzer, "A combinatorial lemma and its application to probability theory" ''Trans. Am. Math. Soc.'' '''82''' (1956) 323-339 {{DOI|10.1090/S0002-9947-1956-0079851-X}} {{ZBL|0071.13003}}</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/b13008070.png" /></td> </tr></table>
+
====Further reading====
 
 
====References====
 
 
<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">[b1]</TD> <TD valign="top">  Guo, Li, "An introduction to Rota-Baxter algebra", Surveys of Modern Mathematics '''4''', International Press (2012) {{ISBN|978-1-57146-253-4}} {{ZBL|1271.16001}} </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>
  
[[Category:TeX partially done]]
+
[[Category:TeX done]]

Latest revision as of 07:40, 18 November 2023

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. A central result in this area is the Spitzer identity $$ \sum_{n=0}^\infty \phi_n(t) \lambda^n = \exp\left[{ \sum_{k=1}^\infty \psi_k(t) \frac{\lambda^k}{k} }\right]\ , $$ 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 remarkable 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 DOI 10.2140/pjm.1960.10.731 Zbl 0095.12705
[a2] G.-C. Rota, "Baxter algebras and combinatorial identities I–II" Bull. Amer. Math. Soc. , 75 (1969) pp. 325–334 DOI 10.1090/S0002-9904-1969-12158-0 Zbl 0319.05008
[a3] G.-C. Rota, "Baxter algebras: an introduction" J.P.S. Kung (ed.) , Gian-Carlo Rota on combinatorics: Introductory papers and commentaries , Birkhäuser (1995) ISBN 3-7643-3713-3 pp. 504–512 Zbl 0841.01031
[a4] Frank Spitzer, "A combinatorial lemma and its application to probability theory" Trans. Am. Math. Soc. 82 (1956) 323-339 DOI 10.1090/S0002-9947-1956-0079851-X Zbl 0071.13003

Further reading

[b1] Guo, Li, "An introduction to Rota-Baxter algebra", Surveys of Modern Mathematics 4, International Press (2012) ISBN 978-1-57146-253-4 Zbl 1271.16001
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