Rota-Baxter Algebra. The Combinatorial Structure of Integral Calculus

Abstract

Gian-Carlo Rota suggested in one of his last articles the problem of developing a theory around the notion of integration algebras, complementary to the already existing theory of differential algebras. This idea was mainly motivated by Rota's deep appreciation for Kuo-Tsai Chen's seminal work on iterated integrals. As a starting point for such a theory of integration algebras Rota proposed to consider a particular operator identity first introduced by the mathematician Glen Baxter. Later it was coined Rota-Baxter identity. In this article we briefly recall basic properties of Rota-Baxter algebras, and present a concise review of recent work with a particular emphasis of noncommutative aspects.

The Pre-Lie Structure of the Time-Ordered Exponential

“…As it turns out Lam's problem amounts to finding a general noncommutative analog of the classical Bohnenblust-Spitzer identity, i.e., identity (18), respectively (20). We therefore find from (16) for weight θ = 0 and denoting ⊲ 0 = ⊲:…”

“…Recall the definition of a unital Rota-Baxter algebra of weight θ ∈ k [4,16,18,38,39]. It is an associative algebra A with unit 1, equipped with a linear endomorphism such that for all x, y ∈ A:…”

“…Especially Cartier contributed to its understanding by using quasi-shuffle like products [27] in the construction of free commutative RB. We refer to [18,39,40] for more details.…”

“…Recall first that a noncommutative shuffle algebra, or dendriform algebra, is an algebra equipped with two products, ↓ and ↑, satisfying the noncommutative version of the Eilenberg-MacLane-Schützenberger axioms of shuffle algebras (see e.g., [18]), that is:…”

“…Following Rota [39,40,41], in recent years the theory of Rota-Baxter algebras of arbitrary weight [4] has emerged as an appropriate setting for studying algebraic and combinatorial aspects underlying the notion of integration. See [18] for a review. Let us mention that in particular we are interested in a more detailed understanding of the classical notion of time-ordered exponential from an algebraic point of view.…”