An algebraic study of multivariable integration and linear substitution

Abstract

We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota–Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota–Baxter hierarchy. We conjecture that the operator relations are a noncommutative Gröbner–Shirshov basis for the ideal they generate.