Conservative algebras of 2-dimensional algebras, IV

Abstract

The notion of conservative algebras appeared in a paper by Kantor in 1972. Later, he defined the conservative algebra W(n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. If n > 1, then the algebra W(n) does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). It looks like W(n) in the theory of conservative algebras plays a similar role to the role of gln in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra W(n) for some n ∈ N. The present paper is part of a series of papers, which is dedicated to the study of the algebra W(2) and its principal subalgebras.