Rota–Baxter algebra
In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map 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 be given. A linear operator R on a k-algebra A is called a Rota–Baxter operator of weight if it satisfies the Rota–Baxter relation of weight :
for all . Then the pair or simply A is called a Rota–Baxter algebra of weight . In some literature, is used in which case the above equation becomes
called the Rota-Baxter equation of weight . The terms Baxter operator algebra and Baxter algebra are also used.
Let be a Rota–Baxter of weight . Then is also a Rota–Baxter operator of weight . Further, for in k, is a Rota-Baxter operator of weight .
Examples
Integration by parts
Integration by parts is an example of a Rota–Baxter algebra of weight 0. Let be the algebra of continuous functions from the real line to the real line. Let : be a continuous function. Define integration as the Rota–Baxter operator
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
In other words
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
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Notes
- ^ Baxter, G. (1960). "An analytic problem whose solution follows from a simple algebraic identity". Pacific J. Math. 10 (3): 731–742. doi:10.2140/pjm.1960.10.731. MR 0119224.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Cartier, P. (1972). "On the structure of free Baxter algebras". Advances in Mathematics. 9 (2): 253–265. doi:10.1016/0001-8708(72)90018-7.
- ^ 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.
- ^ 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.
- ^ Spitzer, F. (1976). "Principles of random walks". Graduate Texts in Mathematics. 34 (Second ed.). New York, Heidelberg: Springer-Verlag.
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(help) - ^ Semenov-Tian-Shansky, M.A. (1983). "What is a classical r-matrix?". Func. Anal. Appl. 17 (4): 259–272. doi:10.1007/BF01076717. S2CID 120134842.
- ^ Connes, A.; Kreimer, D. (2000). "Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem". Comm. Math. Phys. 210 (1): 249–273. arXiv:hep-th/9912092. Bibcode:2000CMaPh.210..249C. doi:10.1007/s002200050779. S2CID 17448874.
- ^ Aguiar, M. (2000). "Infinitesimal Hopf algebras". Contemp. Math. Contemporary Mathematics. 267: 1–29. doi:10.1090/conm/267/04262. ISBN 9780821821268.
- ^ Guo, L.; Keigher, W. (2000). "Baxter algebras and shuffle products". Advances in Mathematics. 150: 117–149. arXiv:math/0407155. doi:10.1006/aima.1999.1858.
External links
- Li Guo. WHAT IS...a Rota-Baxter Algebra? Notices of the AMS, December 2009, Volume 56 Issue 11