Title: Construction of flows of finite-dimensional algebras,
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.