Rota–Baxter algebra

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In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map 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

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Let   be a commutative ring and let   be given. A linear operator   on a  -algebra   is called a Rota–Baxter operator of weight   if it satisfies the Rota–Baxter relation of weight  :

 

for all  . Then the pair   or simply   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  ,   is a Rota-Baxter operator of weight  .

Examples

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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   and  . Then the formula for integration for parts can be written in terms of these variables as

 

In other words

 

which shows that   is a Rota–Baxter algebra of weight 0.

Spitzer identity

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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

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  1. ^ 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.
  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 Mathematics. 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. Vol. 34 (Second ed.). New York, Heidelberg: Springer-Verlag.
  9. ^ 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.
  10. ^ 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.
  11. ^ Aguiar, M. (2000). "Infinitesimal Hopf algebras". Contemp. Math. Contemporary Mathematics. 267: 1–29. doi:10.1090/conm/267/04262. ISBN 9780821821268.
  12. ^ 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.
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