By Johan G. F. Belinfante
During this reprint version, the nature of the booklet, specifically its specialise in classical illustration conception and its computational points, has now not been replaced
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Lie groups at first arose as transformation groups and only later were studied as abstract groups. The shift of emphasis from transformation groups to abstract groups in the Lie theory parallels the corresponding development which took place in the theory of finite groups. The theory of Lie groups, originally called the theory of continuous groups, was developed by Sophus Lie in connection with the problem of integrating systems of differential equations. Thus, groups with analytic structure arose from the study of the solution of differential equations, just as finite group theory arose in the study of algebraic equations.
10 of the technicalities of manifold theory [73]. However, this would definitely entail some loss of generality because there do exist Lie groups which are not isomorphic to any matrix group [33] . To prove that the various classical matrix groups are Lie groups, we must show that they are analytic manifolds and that in each case the group operation is analytic. Let us consider first the general linear groups GL(n, R) and GL(n, C). To see that they are manifolds, we embed them in a Euclidean space and simultaneously erect coordinate systems by noting that the n2 entries of a matrix can be used as the coordinates of a point in a Euclidean space of dimension n2.
The homomorphism induced by a composition of analytic homomorphisms is the composition of their induced homomorphisms. This relation between homomorphisms of Lie groups and Lie algebras can be used to translate many results about Lie groups into related results about their corresponding Lie algebras. For example, the inclusion mapping for a Lie subgroup is an analytic homomorphism, and hence induces the corresponding embedding for their Lie algebras noted above. It is a remarkable fact that any continuous homomorphism of Lie groups is also analytic [124].