Real subalgebras of small dimensions of the matrix Lie algebra \(M(2,\mathbb C)\)
Authors' introduction: ``Real Lie subalgebras of the Lie algebra of square complex matrices of order two arise naturally in the study of affinely homogeneous hypersurfaces in a 3-dimensional complex space. In this connection, one can suggest an approach to the classification of affinely homogeneous surfaces based on the use of such simple objects as algebras of matrices of second order. However, the authors of this paper have not found respective descriptions in the mathematical literature. This paper contains a classification of Lie subalgebras of \(M(2,\mathbb C)\) of all dimensions from 0 to 4, to which the above mentioned problem on homogeneity reduces. On the other hand, analogous subalgebras of all possible dimensions (from 0 to 7) of the Lie algebra \(M(2,\mathbb C)\) may be of interest in other applications. As a special case, we include in this paper the result of the third author [Russ. J. Math. Phys. 10, No. 4, 495--500 (2003; Zbl 1105.81044)], where a complete list of 3-dimensional real Lie subalgebras of \(M(2,\mathbb C)\) is given. Note that, traditionally, 3-dimensional Lie algebras are discussed more often than other. A description of another special case of 2-dimensional subalgebras is given in [\textit{N. S. Pushmina, S. S. Chernykh} and \textit{A. A. Sedaev}, ``Classification of two-dimensional real Lie subalgebras \(M(2,\mathbb C)\),'' in: Proceedings of 5th Intern. Conf. Young Scientists and Students ``Current Problems of Modern Science'' (Samara, 2004), pp. 104--107]. However, in that paper, there are ``superfluous'' algebras and there is no discussion concerning their elimination. In this connection, in Section 2, we list 2-dimensional subalgebras of \(M(2,\mathbb C)\), which will be used in the further discussion, and give necessary comments.'' The remaining cases have been found recently by V. V. Gorbatsevich [Russ. Math. 54, No. 8, 24--28 (2010); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2010, No. 8, 30--35 (2010; Zbl 1227.17009)].
scientific article; zbMATH DE number 5320975