Continuous Evolution Algebras

TL;DR

This paper introduces continuous evolution algebras defined via differentiable matrix functions, explores their relation to ODE solutions and matrix Lie groups, and extends the framework to flow lines on Lie groups.

Contribution

It formulates continuous evolution algebras through differentiable matrices and connects them with matrix Lie groups and flow lines, broadening the theoretical understanding.

Findings

Continuous evolution algebras can be characterized as solutions to certain ODEs.

Matrix Lie groups serve as a natural setting for time-dependent evolution algebras.

Flow lines on matrix Lie groups provide a new perspective on continuous evolution algebras.

Abstract

We formulate the notion of continuous evolution algebra in terms of differentiable matrix-valued functions, to then study those such algebras arising as solutions of ODE problems. Given their dependence on natural bases, matrix Lie groups provide a suitable framework where considering time-variant evolution algebras. We conclude by broadening our approach by considering continuous evolution algebras stemming as flow lines on matrix Lie groups.