ON THREE-DIMENSIONAL COMMUTATIVE ALGEBRAS CONSTRUCTED BY MEANS OF TWO-DIMENSIONAL ALGEBRAS

Abstract

This paper is part of a series of studies devoted to the classification, up to isomorphism, of three-dimensional commutative algebras over an arbitrary base field. The complete classification of finite-dimensional algebras is one of the central problems of algebra theory, yet in the general case it is practically intractable, which is why small dimensions are treated separately. The aim of the study is to construct the list of all three-dimensional commutative algebras containing the two-dimensional algebra A4 (β1,1) as a subalgebra over an arbitrary field F satisfying Char(F)≠2,3 and F2=F. The work employs the apparatus of the matrix of structure constants (MSC) together with the matrix form of isomorphism B=g- Ag⊗2). The equality Aut(A4 (β1,1))={I} substantially narrows the group of basis changes preserving the chosen subalgebra and makes this case essentially different from the previously studied A3 (α1,α4,1-α1 ). By means of a successive normalisation of the parameters and a case-by-case analysis, 30 families of algebras C1,…,C30 are obtained. The scientific novelty of the work is that such a complete list for the subalgebra A4 (β1,1) is presented for the first time. The result obtained serves as a basis for the next stage of the classification - the elimination of redundant families by means of isomorphism invariants.