On Free Baxter Algebras: Completions and the Internal Construction

By Li Guo, William F. Keigher. 2000

100 citations recorded by the source corpus.

We continue the study of free Baxter algebras. There are two goals of this paper. The first goal is to extend the construction of shuffle Baxter algebras to completions of Baxter algebras. This process is motivated by a construction of Cartier and is analogous to the process of completing a polynomial algebra to obtain a power series algebra. However, as we will see later, unlike the close similarity of properties of a polynomial algebra and a power series algebra, properties of a shuffle Baxter algebra and its completion can be quite different.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f833afb4857c1994f0883dbbd73c292653c1dcbf

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Citations in the corpus

Spitzer's identity and the algebraic Birkhoff decomposition in pQFT (2004, background): “(6)To avoid possible confusions, we alert the reader that in [17] the concept of complete filtered RotaBaxter algebras has been defined, where the filtration is canonically derived from the Rota-Baxter operator.”
Grobner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras ∗† (2008, background): “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”
Properties of Free Baxter Algebras (2000, methodology): “Theorem 2.4 [ 9 , 14] Assume that the annihilator of λ ∈ C in the C-module A is zero.”
Mixable shuffles, quasi-shuffles and Hopf algebras (2005, background): “2 KURUSCH EBRAHIMI-FARD AND LI GUO Another paper on a generalization of the shuffle product was published [26] in the same year as the papers [23, 24] on mixable shuffle products.”
On products and duality of binary, quadratic, regular operads (2004, background): “-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]. Date: October 25, 2018. 1They used to be called Baxte”
On Differential Rota-Baxter Algebras (2007, methodology): “The construction of free objects in CDRBλ is carried out in Section 3 (Theorem 3.2) and the construction of free objects in DRBλ is carried out in Section 4 (Theorem 4.2).”
On integro-differential algebras (2012, methodology): “Free commutative Rota-Baxter algebras were obtained in [19, 20] in terms of shuffles and the more general mixable shuffles of tensor powers.”
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”
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.”