# Construction of flows of finite-dimensional algebras

## TL;DR

This paper develops conditions for the existence of flows of finite-dimensional algebras, extending previous work by exploring solutions to the Kolmogorov-Chapman equation with various matrix multiplications and analyzing their dynamics.

## Contribution

It introduces sufficient conditions for solutions to the Kolmogorov-Chapman equation in the context of algebra flows, expanding the class of known finite-dimensional algebra flows.

## Key findings

- Constructed a wide class of algebra flows using matrix exponentials.
- Extended the set of flows with Maksimov's matrix multiplications.
- Derived differential equations describing the dynamics of algebra flows.

## Abstract

Recently, we introduced the notion of flow (depending on time) of finite-dimensional algebras. A flow of algebras (FA) is a particular case of a continuous-time dynamical system whose states are finite-dimensional algebras with (cubic) matrices of structural constants satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are several kinds of multiplications between cubic matrices one has fix a multiplication first and then consider the KCE with respect to the fixed multiplication. The existence of a solution for the KCE provides the existence of an FA. In this paper our aim is to find sufficient conditions on the multiplications under which the corresponding KCE has a solution. Mainly our conditions are given on the algebra of cubic matrices (ACM) considered with respect to a fixed multiplication of cubic matrices. Under some assumptions on the ACM (e.g. power associative, unital, associative, commutative) we describe a wide class of FAs, which contain algebras of arbitrary finite dimension. In particular, adapting the theory of continuous-time Markov processes, we construct a class of FAs given by the matrix exponent of cubic matrices. Moreover, we remarkably extend the set of FAs given with respect to the Maksimov's multiplications of our previous paper (J. Algebra 470 (2017) 263--288). For several FAs we study the time-dependent behavior (dynamics) of the algebras. We derive a system of differential equations for FAs.