On a classification of finite dimensional algebras with respect to the orthogonal (unitary) changes of basis

In this paper, we consider a classification, with respect to the orthogonal (unitary) change of basis, of finite dimensional algebras. A finite system of invariants, which separates nonequivalent algebras, whose systems of structural constants are from an invariant, open, dense set, is given.


Introduction
The classification problem of finite dimensional algebras is one of the important problems in Algebra [1,2]. One of the most used approaches to this problem is the structural (basis free, invariant) method. As an example of such method one can consider a proof of the famous Artin-Wedderburn theorem on classification (semi)simple associative algebras or Cartan's result on classification of (semi)simple Lie algebras. A structural approach to some more general, for example related to Genetics, algebras is not known. If even it can be done one should not expect to have an easy proof and simple formulation of the result. A disadvantage of the structural method is that the classification is understood only with respect to the general linear group.
Another approach to the classification problem is the coordinate (basis based, structural constants) method. In small dimensional cases, for such an approach we refer the reader to [3,4,5]. In case of any finite dimensional algebras, classifications with respect to the general linear groups and some other problems related to them are considered in [6]. In this paper, we also consider the classification problem for any finite dimensional algebras but only with respect to the orthogonal (unitary) change of basis. More exactly, we show how to construct an invariant, open, dense (in the Zariski topology) subset of the space of structural constants, and how to construct a finite system of invariants which separates any two nonequivalent algebras, with respect to the orthogonal (unitary) changes of basis, whose systems of structural constants are in that dense subset. To do it we consider the first the general classification problem of orbits, under one assumption on representation, which may be useful in many other classification problems as well. The main result concerning the classification of finite dimensional algebras (Theorem 2.5) will be derived from the general case though a proof of it may be presented straightforwardly also. vector space over F . Further we consider this representation under the following assumption: Assumption. There exists a nonempty open(in Zariski topology) G-invariant subset V 0 of V and an algebraic map P : V 0 → G such that P (τ (g, v)) = P (v)g −1 whenever v ∈ V 0 and g ∈ G.
This theorem shows that the system of components of τ (P (x), x) is a finite separating system of invariants for the G-orbits In what follows, it is assumed that F is an algebraically closed field.
In particular it is true for g = P (x). But the equality  Proof. We show that the system of components of P (x) is algebraic independent over F (x) GL(m,F ) . Let f (y ij ) i,j=1,2,...,m be any polynomial over F (x) GL(m,F ) and f (P (x)) = 0 which means that j=1,2,...,m by substitution v for x. Therefore for any v ∈ V 1 and g ∈ GL(m, F ) one has 0 = f v (P (τ (g, v))) = f v (P (v)g −1 ).
Let now τ stand for the representation of GL(m, F ) on n = m 3 dimensional vector space where g t stands for the transpose of g.
We will construct an algebraic map P : V 0 → GL(m, F ), for which the equality of the Assumption holds true for any v ∈ V 0 and g ∈ G, in the following way. It is known that where Tr 1 (A) stands for the row vector with entries ∑ n j=1 A j j,i -the contraction on the first upper and lower indices, Tr 2 (A) stands for the row vector with entries ∑ n j=1 A j i,j -the contraction on the first upper and second lower indices.
By induction it is easy to show that (BB t ) k B = g((AA t ) k A)(g −1 ) ⊗2 for any nonnegative integer k. Therefore one has the following row equalities It implies that if P (A) is a matrix, for example, consisting of the first m these rows then P (τ (g, A)) = P (A)g −1 for any g ∈ O(m, F ). So we can state the following result.

Remark 2.7. It is evident that when F is the field of complex numbers Theorem 2.5 and Corollary 2.6 hold true if one changes in them t to * -the transpose with conjugation.
Theorem 2.5 shows that any O(m, F )-invariant function defined on V 0 is a function of the system of components P (x)x(P (x) −1 ) ⊗2 and P (x)P (x) t . But all these components are rational functions of variables x = (x i jk ) i,j,k=1,2,...,m therefore one can ask the following question.