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An introduction to Rota-Baxter algebra. (English) Zbl 1271.16001

Surveys of Modern Mathematics 4. Somerville, MA: International Press; Beijing: Higher Education (ISBN 978-1-57146-253-4/pbk). x, 226 p. (2012).
Let \(R\) be an algebra over a commutative ring \(K\), \(w\) an element of \(K\), \(P\) a linear operator on \(R\). The pair \((R,P)\) is called a Rota-Baxter algebra (or RBA) of weight \(w\) if \(P(x)P(y)=P(xP(y))+P(P(x)y)+wP(xy)\) for all \(x,y\) in \(R\). This generalizes the integration by parts formula (for \(w=0\)). RBA’s originated from a paper of G. Baxter [Pac. J. Math. 10, 731–742 (1960; Zbl 0095.12705)]. Fundamental papers of G.-C. Rota around 1970 brought the subject into the areas of algebra and combinatorics. Survey articles of Rota in the 1990’s helped its renaissance. RBA’s found an important application in the work of A. Connes and D. Kreimer on the renormalization of perturbative quantum field theory [Commun. Math. Phys. 210, No. 1, 249–273 (2000; Zbl 1032.81026); ibid. 216, No. 1, 215–241 (2001; Zbl 1042.81059)].
Another connection to mathematical physics was established by Aguiar that related an RBA of weight 0 to the associative version of the classical Yang-Baxter equation (this Baxter is R. Baxter) [M. Aguiar, Lett. Math. Phys. 54, No. 4, 263–277 (2000; Zbl 1032.17038)]. Aguiar showed that an RBA of weight 0 has the structure of a dendriform algebra, introduced by J.-L. Loday [Enseign. Math., II. Sér. 39, No. 3-4, 269–293 (1993; Zbl 0806.55009)]. A basic example of dendriform algebras is the shuffle product algebra, which was shown to come from RBA’s by K. Ebrahimi-Fard and L. Guo [J. Algebr. Comb. 24, No. 1, 83–101 (2006; Zbl 1103.16025)]. L. Guo and W. Keigher showed that free RBA’s generalized the shuffle product (called the mixable shuffle product) [Adv. Math. 150, No. 1, 117–149 (2000; Zbl 0947.16013)]. M. E. Hoffman introduced another generalization of the shuffle product (called the quasi-shuffle product [J. Algebr. Comb. 11, No. 1, 49–68 (2000; Zbl 0959.16021)]) which plays a role in quantum field theory, and which turns out to be the same as the mixable shuffle product. Subsequently, RBA’s have been connected to operads, Hopf algebras, commutative algebra, combinatorics, number theory and quantum field theory.
Despite all these connections, the theoretical study of RBA’s is still in the early stage of development. The monograph under review organizes these connected results spread out in the literature. This putting together of the basic properties and constructions of RBA’s is aimed at two types of readers. The first type is served by “classical” results that have appeared for several years, and should have had a basic course in abstract algebra. The second type would be interested in further work in this direction. For those, a summary is included at the end of each chapter to discuss more recent developments and to indicate open problems and active areas of current research. The author is well-qualified to do this, having participated heavily in the recent developments. The monograph’s Bibliography lists 26 papers he has authored or coauthored.
The book has three parts, each consisting of two chapters. The first part is on the operator aspects of RBA’s, emphasizing the Rota-Baxter operator \(P\). This includes Spitzer’s identity, which expresses \(\exp(P(w^{-1}\log(1+wrx)))\), for \(r\) in \(R\), as a power series in \(x\) whose coefficients involve iterating \(P\) (this is for weight \(w\) not zero), and a generalization of this identity. There is a discussion of the algebraic Birkhoff decomposition in the Hopf algebra approach of Connes and Kreimer to renormalization of quantum field theory, which is derived from the RBA which is the convolution algebra of the linear maps from a Hopf algebra to a commutative RBA.
The second part constructs various free RBA’s, both commutative and noncommutative. The original construction of free commutative RBA’s was given by Rota and by P. Cartier [Adv. Math. 9, 253–265 (1972; Zbl 0267.60052)]. The book under review constructs free commutative RBA’s in terms of equivalent products of mixable shuffles, quasi-shuffles and shuffle products, with applications to Stirling numbers and partitions. It studies free noncommuative RBA’s on both sets and on algebras in terms of bracketed words and rooted trees.
The third part is on the operad aspect of RBA’s, studying the connection between RBA’s and operad objects, such as dendriform algebras. Free commutative dendriform algebras and tridendriform algebras are compared with free commuatative RBA’s. In the not necessarily commutative context, the author studies the adjoint functor and universal enveloping algebras related to the functor from RBA’s to (tri)dendriform algebras. Finally, in the context of binary quadratic nonsymmetric operads, he considers the relation between Rota-Baxter operads on such operads and the black square product.
We list the titles of the three parts and six chapters of the book. Part I: The Operator Aspect of Rota-Baxter Algebras. Chapter 1: Spitzer’s Identity. Chapter 2: Connected Hopf Algebras and Rota-Baxter Algebras. Part II: The Structure Aspect of Rota-Baxter Algebra. Chapter 3: Free Commutative Rota-Baxter Algebras and Shuffle Products. Chapter 4: Free Noncommutative Rota-Baxter Algebras and Rooted Tress. Part III: The Operad Aspect of Rota-Baxter Algebra. Chapter 5: Rota-Baxter Algebras and Dendriform Algebras. Chapter 6: Rota-Baxter Operators on Operads and Manin Products. There is a Bibliography and an Index.

MSC:

16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16T25 Yang-Baxter equations
17B38 Yang-Baxter equations and Rota-Baxter operators
17A30 Nonassociative algebras satisfying other identities
05E16 Combinatorial aspects of groups and algebras
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16T05 Hopf algebras and their applications
18M60 Operads (general)
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