Gian-Carlo Rota suggested in one of his last articles the problem of developing the notion of integration algebra, complementary to the already existing theory of differential algebras. This idea was mainly motivated by Rota’s deep appreciation for Chen’s fundamental work on iterated integrals. As a starting point for such a theory Rota proposed to consider a particular operator identity first introduced in 1960 by the mathematician Glen Baxter. It was later coined Rota-Baxter identity. Examples range from algebras with a direct decomposition into subalgebras to algebras of functions equipped with the ordinary Riemann integral or its discrete analogs.
Rota-Baxter algebras feature a genuine factorization property. It is intimately related to linear fixpoint equations, such as those, for instance, appearing in the renormalization problem in perturbative quantum field theory. For arbitrary commutative Rota-Baxter algebras, proper exponential solutions of such fixpoint equations are described by what is known as the classical Spitzer identity. The similar classical Bohnenblust-Spitzer identity involves the symmetric group, and generalizes the simple observation that the n-fold iterated integral of a function is proportional to the n-fold product of the primitive of this function. Recently, the seminal Cartier-Rota theory of classical Spitzer-type identities has been generalized to noncommutative Rota-Baxter algebras. Pre-Lie algebras (also known as Vinberg or Gerstenhaber algebras) play a crucial role in this approach. In this talk we will provide a short introduction to Rota-Baxter algebras, and review recent work on Spitzer-type identities.
This talk is based on joint work with Frederic Patras (CNRS, Nice, France) and Dominique Manchon (CNRS, Clermont-Ferrand, France).