Construction of flows of finite-dimensional algebras

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 paper [8]. For several FAs we study the time-dependent behavior (dynamics) of the algebras. We derive a system of differential equations for FAs.