About variety $_{3}\mathbf{N}$ of Leibniz algebras and its subvarieties

T. V. Skoraya ; Yu. Yu. Frolova

Čebyševskij sbornik, Tome 15 (2014) no. 1, pp. 155-185

Résumé

Article represents the review of properties of variety left nilpotent of the class not more than 3 Leibniz algebras and its subvarieties. The characteristic of basic field will be equal to zero. A Leibniz algebra is an algebra with multiplication satisfying the Leibniz identity $(xy)z=(xz)y+x(yz)$. In other words, the operator of right multiplication is a derivation of the algebra. Since Leibniz identity equivalent to the Jacobi identity, in case multiplication in Leibniz algebra is anti-commutative, it is obvious that the Leibniz algebras are generalizations of concept of Lie algebtras. The variety $_{3}\mathbf{N}$ is defined by identity $x(y(zt))\equiv 0$ possesses some extreme properties (properties, which any its own subvariety possesses, while the variety doesn't possess them). As the basic field has zero characteristic zero, then any identity is equivalent to the system of multilinear identities, that allows to use well-developed theory of representations of the symmetric group. In addition to using the classical results of the structural theory of rings and linear algebras, representation theory, as well as the structural theory of varieties of associative algebras, and the use of original asymptotic and combinatorial arguments with application identities and Young diagrams allowed to receive the following results: the variety $_{3}\mathbf{N}$ has almost exponential growth, almost polynomial growth of colength, almost finite multiplicity. Moreover, this variety has almost associative type, that is his own cocharacter any subvarieties lies in the hook. In this work are considered also subvarieties of variety $_{3}\mathbf{N}$: held description of the complete list of varieties with almost polynomial growth; proved integrality of exponents any proper subvariety of variety $_{3}\mathbf{N}$.

Détail

Keywords: varieties of linear algebras, numerical characteristics of varieties, growth of variety, multiplicity of variety, colength of variety, variety with almost polynomial growth, variety with almost associative type, exponent of variety, Leibniz algebras.