Generalized derivations and automorphisms of some classes of algebras

This thesis focus on the problem of generalized derivations of some classes of al- gebras over complex field. It defined the concept of the generalization via some complex parameters, and in particular on some values of the parameters. It uses an algorithm in the computation of the generalized derivations of some algebras of lower dimensional cases. Particularly, the associative algebras and the Leibniz al- gebras. The results of the computations are presented in a matrix form, and further interpreted. Thus, different subalgebras, subspaces, and two one-parametric sets of linear operators are obtained. Intersection among various subspaces are also found. The classification result of the generalized derivation of algebras are given. Eight different subspaces and their structures are found in the case of associative algebra, which includes, the classical derivation. In the Leibniz algebra case, we had eight subspaces too, with different structures, it includes two one-parametric sets as well. Furthermore, by using the two one-parametric sets, two invariant functions are de- fined. The functions together with some criteria are used to establish contractions among algebras of lower dimensions. The list of contractions among the Leibniz algebras and the associative dialgebras of dimensions 2 and 3 are given. The work also compute the automorphism group of the Leibniz and the associative dialgebras. The work also described the concept of a generalized automorphism of algebras. The concept is defined with the aid of some sets of automorphisms. It is found out that, there exists an isomorphism between some corresponding sets of generalized deriva- tions and the generalized automorphisms. In the same way, we found an isomor- phism among various intersections of the subspaces of the generalized derivations with that of the generalized automorphisms. The inner derivation of algebras is also looked into. As a result, additional invariant characteristic of algebras are found.

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