Rota–Baxter algebra

In mathematics, a Rota–Baxter algebra is an algebra, usually over a field k, together with a particular k-linear map R which satisfies the weight-θ 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]

Definition and first properties

Let A be a k-algebra with a k-linear map R on A and let θ be a fixed parameter in k. We call A a Rota-Baxter k-algebra and R a Rota-Baxter operator of weight θ if the operator R satisfies the following Rota–Baxter relation of weight θ:

$R(x)R(y) + \theta R(xy) = R(R(x)y + xR(y)).\,$

The operator R:= θ id − R also satisfies the Rota–Baxter relation of weight θ.

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 operator

$I(f)(x) = \int_0^x f(t) dt \;.$

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 as

$F(x)G(x) = \int_0^x f(t) G(t) dt + \int_0^x F(t)g(t) dt \;.$

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.

See also

• Dendriform algebra

Notes

1. Baxter, G. (1960). "An analytic problem whose solution follows from a simple algebraic identity". Pacific J. Math. 10: 731–742. doi:10.2140/pjm.1960.10.731. MR 0119224.
2. Rota, G.-C. (1969). "Baxter algebras and combinatorial identities, I, II". Bull. Amer. Math. Soc. 75 (2): 325–329. doi:10.1090/S0002-9904-1969-12156-7.; 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. Cartier, P. (1972). "On the structure of free Baxter algebras". Advances in Math. 9 (2): 253–265. doi:10.1016/0001-8708(72)90018-7.
6. Atkinson, F. V. (1963). "Some aspects of Baxter's functional equation". J. Math. Anal. Appl. 7: 1–30. doi:10.1016/0022-247X(63)90075-1.
7. Spitzer, F. (1956). "A combinatorial lemma and its application to probability theory". Trans. Amer. Math. Soc. 82 (2): 323–339. doi:10.1090/S0002-9947-1956-0079851-X.
8. Spitzer, F. (1976). "Principles of random walks". Graduate Texts in Mathematics 34 (Second ed.). New York, Heidelberg: Springer-Verlag.

External links

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