Abstract
This chapter discusses some basic concepts. One of the fundamental problems of the theory of Lie algebras is to determine all nonisomorphic Lie algebras. If two Lie algebras are isomorphic, then in regard to a suitable basis, they have the same structure constants. Conversely, Lie algebras with the same set of structure constants are isomorphic. When M is nonsingular and symmetric, orthogonal algebra is obtained. Because any complex nonsingular symmetric matrix is congruent to the identity matrix, the orthogonal algebra, in regard to some M, can be considered as consisting of all skew-symmetric matrices. There are two series of orthogonal Lie algebras. One-dimensional Lie algebras are simple and any abelian Lie algebra of dimension greater than one is not simple. Except for the one-dimensional Lie algebra, simple Lie algebras are not abelian.
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