A Study about One Representation of the Hamiltonian Complex finite dimensional Lie Algebra - Taffach, Nader Mahmoud (Arab Journal for Sciences and Research Publishing, 2022)
The importance of group theory and its classifications lies in many engineering, physical and chemical fields, and in particular in concepts related to the concept of symmetry. In this article, we study an issue related to the representation of Lie algebra that is closely related to the topic of Lie groups and their representations. The construction of a complex Lie algebra generates a correspondence between complete connect Lie groups with a trivial center and semi-simple Lie complex algebras. In this article, we found a harmonic representation of the finite-dimension complex Lie algebra in terms of its base elements and an adjoint matrix for an arbitrary element of it. This article consists of an introduction and two main sections. In the first section, the basic definitions and concepts that will be relied upon were mentioned in the second important section of the article, in which a basic theorem with its proof will be presented, through this theorem any complex Lie algebra of a finite dimension can be represented in terms of its base elements and an adjoint matrix for an element of it.
Keywords: algebra over field; Lie algebra; complex Lie algebra; adjoint matrix; representation of Lie algebra