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 $\lambda$ be given. A linear operator $R$ on a $k$ -algebra $A$ is called a Rota–Baxter operator of weight $\lambda$ if it satisfies the Rota–Baxter relation of weight $\lambda$ :

R(x)R(y)=R(R(x)y)+R(xR(y))+\lambda R(xy)

for all $x,y\in A$ . Then the pair $(A,R)$ or simply $A$ is called a Rota–Baxter algebra of weight $\lambda$ . In some literature, $\theta =-\lambda$ is used in which case the above equation becomes

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

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.

Bohnenblust–Spitzer identity

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. Vol. 34 (Second ed.). New York, Heidelberg: Springer-Verlag.
^ 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

[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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