Exponential series without denominators

TL;DR

This paper explores a new form of the exponential series in Zinbiel and dendriform algebras, revealing novel functional equations and connections to brace products and pre-Lie products.

Contribution

It introduces a denominator-free exponential series for Zinbiel algebras and extends it to dendriform algebras, establishing new functional equations and algebraic relationships.

Findings

Exponential series can be expressed without denominators in Zinbiel algebras.

The series satisfies a functional equation similar to Baker-Campbell-Hausdorff in dendriform algebras.

Obstruction series equals the sum of brace products; iterated pre-Lie product replaces Eulerian idempotent in multilinear case.

Abstract

For a commutative algebra which comes from a Zinbiel algebra the exponential series can be written without denominators. When lifted to dendriform algebras this new series satisfies a functional equation analogous to the Baker-Campbell-Hausdorff formula. We make it explicit by showing that the obstruction series is the sum of the brace products. In the multilinear case we show that the role the Eulerian idempotent is played by the iterated pre-Lie product.