On Differential Rota-Baxter Algebras

By Li Guo, William F. Keigher. 2007

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Abstract A Rota–Baxter operator of weight λ is an abstraction of both the integral operator (when λ = 0 ) and the summation operator (when λ = 1 ). We similarly define a differential operator of weight λ that includes both the differential operator (when λ = 0 ) and the difference operator (when λ = 1 ). We further consider an algebraic structure with both a differential operator of weight λ and a Rota–Baxter operator of weight λ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objects in the corresponding categories. In the commutative case, the free objects are given in terms of generalized shuffles, called mixable shuffles. In the noncommutative case, the free objects are given in terms of angularly decorated rooted forests. As a byproduct, we obtain structures of a differential algebra on decorated and undecorated planar rooted forests.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/61b6b3e1524aaddfcad2aa7354cec37812e608ea

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Citations in the corpus

Operated semigroups, Motzkin paths and rooted trees (2007, methodology): “Given the recent progresses on Rota-Baxter algebra in both theoretical and application aspects [1, 2, 10, 16, 17, 19, 20, 26, 27, 29 , 30, 31], it is desirable to obtain convenient constructions of free Rota-Baxter algebras.”
Splitting of operations, Manin products and Rota-Baxter operators (2011, background): “On the other hand, the well-known Koszul dual ity of Manin black product suggests a possible duality for the successors and Rota-Baxter operators, maybe to di fferential type operators [5, 22, 23].”
On the operad of associative algebras with derivation (2009, background): “Since, in the relations, the variables stay in the same order, the category of associative algebras with derivation can be encoded by a nonsymmetric operad that we denote by AsDer.”
On integro-differential algebras (2012, methodology): “We also review the following construction from [21].”
Grobner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras ∗† (2008, methodology): “Guo and Keigher [54] introduced a notion of λ-differential Rota–Baxter algebra which is a Rota–Baxter K -algebra of weight λwith a λ-differential operator D such that DP = IdR.”