The structure theory of Lie algebras have been deeply investigated for many years by mathematicians, namely algebraists and specialists in representation theory, and by theoretical physicists. Leibniz algebras present a "non commutative" analogue of Lie algebras and they were studied by French mathematician J.-L. Loday, as algebras which satisfy the following identity:
[x,[y,z]]=[[x,y],z]-[[x,z],y].
In studying the properties of the homology of Lie algebras Loday noted that if in the definition of an n-th chain the exterior product is changed by the tensor product then in order to prove the derived property defined on chains it is sufficient that the algebra satisfies the Leibniz identity instead of antisymmetricity and Jacobi identities. This motivated the introduction of Leibniz algebras, which are a "non skew-symmetric" generalization of Lie algebras.
In this talk we will discuss on properties of nilpotent Leibniz algebras and we present the main properties of their derivations. The application of derivations of nilpotent Leibniz algebras in the description of solvable Leibniz algebras will be shown. In addition, some open problems will be highlighted.
Bakhrom Omirov is a professor at the National University of Uzbekistan. His research focused on non-associative algebras and superalgebras. In particular, he is one of the authors of the monograph devoted to the structure theory of Leibniz algebras. Bakhrom Omirov is a member of The World Academy of Sciences, which includes 66 countries), as well as Uzbek and American Societies.
He is a winner of several prestigious fellowships (Fulbright, USA; INTAS, Belgium).
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