Conservative algebras of $2$-dimensional algebras, V

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

m a t h f r a k {g l}_{n}

in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra

W (n)

for some

n i n m a t h b b N .

The present paper is a part of a series of papers, which dedicated to the study of the algebra

W (2)

and its principal subalgebras.

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