By Johan G. F. Belinfante
During this reprint version, the nature of the booklet, specifically its specialize in classical illustration conception and its computational elements, has no longer been replaced
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Extra info for A Survey of Lie Groups and Lie Algebra with Applications and Computational Methods
For the group GL(n, C), we treat real and imaginary parts of the matrix entries as separate coordinates and so use a real Euclidean space of dimension 2n2 rather than a complex space of dimension n2. Since the determinant is a continuous function in these coordinates, all points sufficiently near a point representing a nonsingular matrix also represent nonsingular matrices. Hence the general linear group is an open subset of Euclidean space, and the Euclidean coordinates may be used as curvilinear coordinates for the general linear group.
The set of all such ordered pairs can be regarded as the basis of an infinite-dimensional vector space consisting of all their finite formal linear combinations. A typical vector in this space is a vector Consider the set of all vectors in this space of the forms and The quotient space of the whole infinite-dimensional space with respect to the infinite-dimensional subspace spanned by these vectors is called the tensor product vector space Vt ® V2 . In this construction, the elements of the vector space V1 ® V2, which we may naturally call tensors, are of course cosets of vectors in the original space with respect to the subspace.
It is clear that contraction very much resembles the operation of taking a trace. Indeed, they become the same when the spaces V* ® Fand linF (V) are identified in a certain natural way. The process of complexification for a real Lie algebra can be explained as follows. We may recall that a real or complex vector space with a bilinear vector multiplication satisfying the properties and the Jacobi identity is called a Lie algebra. If e ^ , • • • , en is a basis for a real Lie algebra L, then the Lie product is determined by the structure constants cy* defined by Since the definition of a Lie algebra amounts to imposing restrictions on the structure constants, any real Lie algebra L with structure constants c^ can be extended to a complex Lie algebra having the same structure constants.
A Survey of Lie Groups and Lie Algebra with Applications and Computational Methods by Johan G. F. Belinfante