The Derivation Algebras Of Some Leibniz Algebras And Their Properties

In [1], [2] several classes of new algebras were introduced. Some of them have two generating operations and they are called dialgebras.The first motivation to introduce such algebraic structures was problems in algebraic K-theory and they are related with well known Lie and associative algebras.The categories of algebras over these operads assemble into a commutative diagram of functors which reflects the Koszul duality.Leibniz algebras present a "non commutative" analogue of Lie algebras and they were introduced by J.-L. Loday[3], as algebras which satisfy the following identity:They appeared to be related in a natural way to several topics such as differential geometry,homological algebra, classic algebraic topology, algebraic K-theory, loop spaces, noncommutative geometry, etc.Basing on the existed characteristic low dimensional algebras, this paper makes use of the properties of Leibniz algebras determined the properties of some nilpotent non-Lie Leibnizalgebras. This paper includes four parts. In the first part, the history and some important results of Leibniz algebras are introduced. In the second part, we calculate derivation algebrasof all 4-dimensional nilpotent complex Leibniz algebras and determine the solvability of the derivation algebras, and then prove that the derivation algebras of all nilpotent Leibniz algebras are solvable. In the third part, based on the existed 4-dimensional nilpotent complexLeibniz algebras and some special derivation algebras, we extend a new type nilpotent complex Leibniz algebras. In the last part, we make use of Matlab programs to calculate the second cohomology of the simplest filiform Lie algebras and the second cohomology of the Leibniz algebras.