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By Johan G. F. Belinfante

During this reprint version, the nature of the ebook, in particular its specialise in classical illustration concept and its computational facets, has now not been replaced

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Many of the ideas here are based on the more general theory of topological groups, but for Lie groups some simplifications occur. Lie groups, being manifolds, are locally homeomorphic to open regions of Euclidean space, and therefore have very nice local properties. In particular, every Lie group is a locally compact normal Hausdorff space. A topological space is connected if it has no proper subsets which are both open and closed. An arc y in a topological space is a continuous mapping from the closed interval [0,1] into the space, and a loop is an arc for which y(0) = y(l).

Equivalently, the sum of two subspaces may be described as the subspace spanned by their union. The set of all subspaces of a vector space is said to be a lattice under the two operations D and +. This means that these operations satisfy certain axioms somewhat reminiscent of Boolean algebra, but not quite as strong [112], [134]. In particular, the distributive laws between fl and + do not hold, and there is no analogue of the de Morgan laws of complementation. 21 The Lattice of Ideals of a Lie Algebra 43 every pair of elements has a least upper bound and a greatest lower bound.

The requirement that the quantization map be a Lie algebra homomorphism does not define it uniquely on the whole set of dynamical variables. 13 have to be removed by additional assumptions [108], [250], [251]. In the last analysis, of course, there is no reason to expect that any quantization procedure will be entirely satisfactory for physics. Strictly speaking, the correspondence principle only requires that the classical limit of quantum mechanics be derivable in some precise way, but not necessarily the other way around.

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