Open AccessThe classification of three-dimensional Lie algebras on complex field
Author(s) -
E. R. Shamardina
Publication year - 2020
Publication title -
differencialʹnaâ geometriâ mnogoobrazij figur
Language(s) - English
Resource type - Journals
eISSN - 2782-3229
pISSN - 0321-4796
DOI - 10.5922/0321-4796-2020-51-16
Subject(s) - mathematics , pure mathematics , isomorphism (crystallography) , lie algebra , abelian group , field (mathematics) , algebra over a field , chemistry , crystal structure , crystallography
In this paper, we study the classification of three-dimensional Lie algebras over a field of complex numbers up to isomorphism. The proposed classification is based on the consideration of objects invariant with respect to isomorphism, namely such quantities as the derivative of a subalgebra and the center of a Lie algebra. The above classification is distinguished from others by a more detailed and simple presentation. Any two abelian Lie algebras of the same dimension over the same field are isomorphic, so we understand them completely, and from now on we shall only consider non-abelian Lie algebras. Six classes of three-dimensional Lie algebras not isomorphic to each other over a field of complex numbers are presented. In each of the classes, its properties are described, as well as structural equations defining each of the Lie algebras. One of the reasons for considering these low dimensional Lie algebras that they often occur as subalgebras of large Lie algebras