Rota–Baxter algebra explained

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

k be a commutative ring and let λ be given. A linear operator R on a k -algebra A is called a Rota–Baxter operator of weight λ if it satisfies the Rota–Baxter relation of weight λ :R(x)R(y) = R(R(x)y) + R(xR(y)) + λ R(xy)

for all

x, y \in A . Then the pair (A,R) or simply A is called a Rota–Baxter algebra of weight λ . In some literature, \theta =-λ is used in which case the above equation becomesR(x)R(y) + \theta R(xy) = R(R(x)y) + R(xR(y)),

called the Rota-Baxter equation of weight

\theta . The terms Baxter operator algebra and Baxter algebra are also used.

Let

R be a Rota–Baxter of weight λ . Then -λ Id - R is also a Rota–Baxter operator of weight λ . Further, for \mu in k , \mu R is a Rota-Baxter operator of weight \mu λ .

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 operatorI(f)(x) = \int ^{x} _{0} 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 asF(x)G(x) = \int ^{x} _{0} f(t) G(t) dt + \int ^{x} _{0} 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.

Bohnenblust - Spitzer identity

External links

• Li Guo. WHAT IS...a Rota-Baxter Algebra? Notices of the AMS, December 2009, Volume 56 Issue 11

Notes and References

1. G. . Baxter . An analytic problem whose solution follows from a simple algebraic identity . Pacific J. Math. . 10 . 3. 731–742 . 1960 . 0119224 . 10.2140/pjm.1960.10.731. free .
2. G.-C. . Rota . Baxter algebras and combinatorial identities, I, II . Bull. Amer. Math. Soc. . 75 . 325–329 . 1969 . 10.1090/S0002-9904-1969-12156-7 . 2 . free .

   ibid. 75, 330 - 334, (1969). Reprinted in: Gian-Carlo Rota on Combinatorics: Introductory papers and commentaries, J. P. S. Kung Ed., Contemp. Mathematicians, Birkhäuser Boston, Boston, MA, 1995.

3. G.-C. Rota, Baxter operators, an introduction, In: Gian-Carlo Rota on Combinatorics, Introductory papers and commentaries, J.P.S. Kung Ed., Contemp. Mathematicians, Birkhäuser Boston, Boston, MA, 1995.
4. G.-C. Rota and D. Smith, Fluctuation theory and Baxter algebras, Instituto Nazionale di Alta Matematica, IX, 179 - 201, (1972). Reprinted in: Gian-Carlo Rota on Combinatorics: Introductory papers and commentaries, J. P. S. Kung Ed., Contemp. Mathematicians, Birkhäuser Boston, Boston, MA, 1995.
5. P. . Cartier . On the structure of free Baxter algebras . . 9 . 2. 253–265 . 1972 . 10.1016/0001-8708(72)90018-7 . free .
6. F. V. . Atkinson . Some aspects of Baxter's functional equation . J. Math. Anal. Appl. . 7 . 1–30 . 1963 . 10.1016/0022-247X(63)90075-1 . free .
7. F. . Spitzer . A combinatorial lemma and its application to probability theory . Trans. Amer. Math. Soc. . 82 . 2. 323–339 . 1956 . 10.1090/S0002-9947-1956-0079851-X . free .
8. Book: Spitzer, F. . Principles of random walks . Second . Graduate Texts in Mathematics . 34 . Springer-Verlag . New York, Heidelberg . 1976 .
9. M.A. . Semenov-Tian-Shansky . What is a classical r-matrix? . Func. Anal. Appl. . 17 . 4. 259–272 . 1983 . 10.1007/BF01076717 . 120134842 .
10. Connes. A.. Kreimer. D.. Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Comm. Math. Phys.. 2000. 210. 1. 249–273. 10.1007/s002200050779. hep-th/9912092. 2000CMaPh.210..249C . 17448874 .
11. M. . Aguiar . Infinitesimal Hopf algebras . Contemp. Math. . 267 . 1–29 . 2000 . 10.1090/conm/267/04262 . Contemporary Mathematics . 9780821821268 .
12. Guo. L.. Keigher. W.. Baxter algebras and shuffle products. Advances in Mathematics. 2000. 150. 117–149. 10.1006/aima.1999.1858. free. math/0407155.