Conservative algebras of 2-dimensional algebras, II

Abstract

), Yury Volkov (wolf86 666@list.ru).Abstract. In 1990 Kantor 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). We describe automorphisms, one-sided ideals, and idempotents of W (2). Also similar problems are solved for the algebra W 2 of all commutative algebras on the 2-dimensional vector space and for the algebra S 2 of all commutative algebras with trace zero multiplication on the 2-dimensional vector space.

Conservative algebras of 2-dimensional algebras, III

“…in this basis is given in [15]. In this work we use another basis for the algebra W (2) (from [16]). Let introduce the notation…”

“…In 2017 Kaygorodov and Volkov [16] described automorphisms, one-sided ideals, and idempotents of W (2). Also a similar problem is solved for the algebra W 2 of all commutative algebras on the 2-dimensional vector space and for the algebra S 2 of all commutative algebras with zero multiplication trace on the 2-dimensional vector space.…”

Conservative algebras and superalgebras: a survey

“…Therefore, different nonzero vectors a give rise to isomorphic algebras, which we denote by U(V ). In particular (see [22]), we have an injective homomorphism of a GL(V, a) = {ϕ ∈ GL(V ) : ϕ(a) = a} to Aut(U(V ), a ). If V = V n is a finite-dimensional space of dimension n, then we denote U(V ) by U(n).…”

“…Properties of the algebra U(2) were studied in various articles. For example, in the paper [20] the authors described the derivations and subalgebras of codimension one of U(2) and its simple terminal subalgebras W 2 , S 2 (see in the next section), and in the article [22] the one-sided ideals, automorphisms and idempotents of U(2) were described. Note that by definition the classification of idempotents of (U(2), u ) corresponds to the classification of 2-dimensional algebras with u as a left quasiunit.…”

“…• It would also be interesting to study the algebraic properties of universal conservative algebras U(n) for n > 2, continuing the work of the papers [20], [22] (that is, to find one-sided ideals, automorphisms, maximal subalgebras, etc. of these algebras)…”