A power matrix approach to the Witt algebra and Loewner equations

TL;DR

This paper introduces a power matrix framework for the Witt algebra and Loewner equations, establishing new links between formal power series, matrix exponentials, and holomorphicity properties.

Contribution

It develops a novel power matrix approach to formal power series and Loewner equations, connecting matrix exponentials with the group of power matrices and holomorphicity.

Findings

Matrix exponential is surjective onto the group of power matrices

Coefficients of solutions are entire functions of infinitesimal generator coefficients

Holomorphicity of generators relates to holomorphicity of exponentiated matrices

Abstract

The theory of formal power series and derivation is developed from the point of view of the power matrix. A Loewner equation for formal power series is introduced. We then show that the matrix exponential is surjective onto the group of power matrices, and the coefficients are entire functions of finitely many coefficients of the infinitesimal generator. Furthermore coefficients of the solution to the Loewner equation with constant infinitesimal generator can be obtained by exponentiating an infinitesimal power matrix. We also use the formal Loewner equations to investigate the relation between holomorphicity of an infinitesimal generator to holomorphicity of the exponentiated matrix.