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 ${\mathfrak{gl}}_n$ in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra $W(n)$ for some $n\in \mathbb{N}.$ The present paper is a part of a series of papers, dedicated to the study of the algebra $W(2)$ and its principal subalgebras.

References in corpus (12)

• The variety of $2$-dimensional algebras over an algebraically closed field
• Degenerations of nilpotent algebras
• The algebraic and geometric classification of nilpotent terminal algebras
• Non-associative central extensions of null-filiform associative algebras
• The classification of $2$-dimensional rigid algebras
• The variety of dual mock-Lie algebras
• Conservative algebras of 2-dimensional algebras
• Conservative algebras of $2$-dimensional algebras, II
• One-generated nilpotent terminal algebras
• On the Kantor product, II
• The universal conservative superalgebra
• On The Kantor Product