On the bracket of integrable derivations

TL;DR

This paper demonstrates that multi-variate Hasse-Schmidt derivations can be decomposed into simpler components and shows that the property of m-integrability is preserved under the derivation bracket operation.

Contribution

It introduces a decomposition method for multi-variate Hasse-Schmidt derivations and proves the closure of m-integrable derivations under the bracket operation.

Findings

Decomposition of multi-variate Hasse-Schmidt derivations into substitution maps and uni-variate derivations.

Proof that the bracket of two m-integrable derivations remains m-integrable.

Extension of integrability results to both finite and infinite cases.

Abstract

We prove that any multi-variate Hasse-Schmidt derivation can be decomposed in terms of substitution maps and uni-variate Hasse-Schmidt derivations. As a consequence we prove that the bracket of two $m$-integrable derivations is also $m$-integrable, for $m$ a positive integer or infinity.