Subjects of research: Finite dimensional complex Zinbiel algebras, filiform Leibniz algebras.
Purpose of work: To investigate n-dimensional complex filiform Zinbiel algebras, to examine the structural theory of Zinbiel algebras.
Methods of research: In this work methods of structural constants, classification methods, gradation methods and the methods of invariant theory are used.
The results obtained and their novelty: The main results of the work are the following:
- criteria of isomorphism of filiform Leibniz algebras class, natural gradation of which are Lie algebras, is obtained;
- the classification of four-dimensional complex Zinbiel algebras is obtained;
- zero-filiform and filiform complex Zinbiel algebras are described. Based on this description, the derivations of such algebras are investigated. Moreover, the description was extended to the class of complex naturally graded quasi-filiform Zinbiel algebras;
- some properties of characteristic sequence for the Zinbiel algebras are obtained. Furthermore, the classifications of n-dimensional complex Zinbiel algebras with nilindex n-2 with characteristic sequences (n-3, 3) and (n-3, 1, 1, 1) are obtained.
Practical value: The results of the dissertation are of theoretical character.
Degree of embed and economic effectivity: It can be used at reading of special courses.
Field of application: The main scientific results and methods presented in the work can be used in research of other algebras and superalgebras, in the theory of categories, in the study of algebras with various types of gradation, in calculation of cohomological and homological groups.
