Citation
Adamu, Abdulkadir
(2019)
Generalized derivations and automorphisms of some classes of algebras.
Doctoral thesis, Universiti Putra Malaysia.
Abstract
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 oneparametric 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 oneparametric sets as well. Furthermore, by using the two
oneparametric 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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