Rota–Baxter algebra

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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[edit]

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[edit]

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[edit]

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[edit]

See also[edit]

Notes[edit]

  1. ^ Baxter, G. (1960). "An analytic problem whose solution follows from a simple algebraic identity". Pacific J. Math. 10: 731–742. 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[edit]