Approximation of an algebra by evolution algebras

Abstract: It is known that any multiplication of a finite dimensional algebra is determined by a matrix of structural constants. In general, this is a cubic matrix. Difficulty of investigation of an algebra depends on the cubic matrix. Such a cubic matrix defines a quadratic mapping called an evolution operator. In the case of evolution algebras, the cubic matrix consists many zeros allowing to reduce it to a square matrix. In this paper for any finite dimensional algebra we construct a family of evolution algebras corresponding to Jacobian of the evolution operator at a point of the algebra. We obtain some results answering how properties of an algebra depends on the properties of the corresponding family of evolution algebras. Moreover, we consider evolution algebras corresponding to 2 and 3-dimensional nilpotent Leibniz algebras. We prove that such evolution algebras are nilpotent too. Also we classify such evolution algebras.