Baxter Algebras and Shuffle Products

By Li Guo, William F. Keigher. 2000

242 citations recorded by the source corpus.

In this paper we generalize the well-known construction of shuffle product algebras by using mixable shuffles, and prove that any free Baxter algebra is isomorphic to a mixable shuffle product algebra. This gives an explicit construction of the free Baxter algebra, extending the work of Rota and Cartier.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/781b6e70a20465b0c08d650d62530b85220b9b22

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Citations in the corpus

Rota–Baxter algebras and dendriform algebras (2005, background): “Recall [41, 42] that a free nonunitary Rota-Baxter algebra over B is a nonunitary Rota-Baxter algebra X (B) with a Rota-Baxter operator RB and a nonunitary algebra homomorphism jB : B → X NC, (B) such that, for any nonunitary Rota-Baxter algebra A and any nonunitary algebra homomorphism f : B → A, t”
On integro-differential algebras (2012, methodology): “n≥0 A T is the mixable shuffle algebra [17, 19, 24] on the k-algebra AT .”
Grobner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras ∗† (2008, methodology): “A differential algebra R of weight λ, called λ-differential algebra, is an associative K -algebra with linear λ-differential operator D(x) such that D(xy) = D(x)y+ xD(y)+ λD(x)D(y), x, y ∈ R. Guo and Keigher [54] introduced a notion of λ-differential Rota–Baxter algebra which is a Rota–Baxter K -alg”
Deformations of associative Rota-Baxter operators (2019, background): “In particular, the identity of the Rota-Baxter operator can be considered as a possible generalization of the standard shuffle relation [19].”
Spitzer's identity and the algebraic Birkhoff decomposition in pQFT (2004, background): “Keigher gave a very concise description of the latter in terms of a mixable shuffle product [16, 17], which provides a generalization of the classical shuffle product [13].”
Properties of Free Baxter Algebras (2000, methodology): “The explicit descriptions of free Baxter algebras obtained in [ 8 , 9] enable us to consider this question for a more general class of Baxter algebras.”
Operated semigroups, Motzkin paths and rooted trees (2007, background): “A Rota-Baxter algebra (of weight λ) [4, 7, 15, 26, 36, 37] is defined to be an associative algebra A with a linear operator P such that P(x)P (y) = P(xP (y)) + P(P (x)y) + λP (xy), ∀x, y ∈ A; (1)”
FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES (2005, methodology): “In fact, such a relation for Lie algebras was introduced independently by Belavin and Drinfeld [8], and Semenov-Tian-Shansky [58] in the 1980s, under the disguise of r-matrices, of the (modified) classical Yang–Baxter equation, named after the physicists Chen-ning Yang and Rodney Baxter.”
On Free Baxter Algebras: Completions and the Internal Construction (2000, methodology): “Let Rings denote the category of commutative rings with identity.”
Loday-Type Algebras and the Rota–Baxter Relation (2002, methodology): “They appeared for the first time in the work of the mathematician G. Baxter [3] and were then intensively studied by F. V. Atkinson [4], J. B. Miller [5], G.-C. Rota [6, 7], P. Cartier [8] and others, while more recently they reappeared in the work of L. Guo [2].”
Baxter Algebras and Hopf Algebras (2003, background): “The C-algebra C has a natural structure of a C-coalgebra with ∆C : C → C ⊗ C, c 7→ c⊗ 1, c ∈ C and εC = idC : C → C.”
On Differential Rota-Baxter Algebras (2007, methodology): “The forgetful functor from the category DRBλ to the category Set of sets is the composition of the forgetful functors fromDRBλ toAlg and fromAlg to Set.”
On products and duality of binary, quadratic, regular operads (2004, unknown): “…inspired by the connection [1, 10, 27] with Rota-Baxter operators1 which were introduced by G. Baxter [6] in 1960 and were actively studied in the 1970s [43, 44] and again in recent years in connection with several areas of mathematics and physics [5, 8, 9, 11, 12, 15, 15, 23, 24, 21, 22, 25].”
On the algebra of quasi-shuffles (2005, background): “(a) From Proposition 1.7 it is clear that the quasi-shuffle algebra over a commutative ring is a CTD-algebra.”