Flow of finite-dimensional algebras

TL;DR

This paper introduces the concept of flows of finite-dimensional algebras as continuous-time dynamical systems governed by Kolmogorov-Chapman equations, exploring their properties, stochastic variants, and applications to algebraic structures.

Contribution

It formalizes the notion of algebra flows, connects them with Kolmogorov-Chapman equations, and constructs examples including stochastic and periodic flows with diverse algebraic properties.

Findings

Defined stochastic flows of algebras generated by quadratic stochastic processes.

Reduced Kolmogorov-Chapman equations for certain cubic matrices to square matrix equations.

Constructed a continuum of algebras for periodic flows and demonstrated density in the set.

Abstract

Each finite-dimensional algebra can be identified to the cubic matrix given by structural constants defining the multiplication between the basis elements of the algebra. In this paper we introduce the notion of flow (depending on time) of finite-dimensional algebras. This flow can be considered as a particular case of (continuous-time) dynamical system whose states are finite-dimensional algebras with matrices of structural constants satisfying an analogue of Kolmogorov-Chapman equation. These flows of algebras (FA) can also be considered as deformations of algebras with the rule (the evolution equation) given by Kolmogorov-Chapman equation. We mainly use the multiplications of cubic matrices which were introduced by Maksimov and consider Kolmogorov-Chapman equation with respect to these multiplications. If all cubic matrices of structural constants are stochastic (there are several kinds of stochasticity) then the corresponding FA is called stochastic FA (SFA). We define SFA generated by known quadratic stochastic processes. For some multiplications of cubic matrices we reduce Kolmogorov-Chapman equation given for cubic matrices to the equation given for square matrices. Using this result many FAs are given (time homogenous, time non-homogenous, periodic, etc.). For a periodic FA we construct a continuum set of finite-dimensional algebras and show that the corresponding discrete time FA is dense in the set. Moreover, we give a construction of an FA which contains algebras of arbitrary (finite) dimension. For several FAs we describe the time depending behavior (dynamics) of the properties to be baric, limit algebras, commutative, evolution algebras or associative algebras.