Theory of Lie Groups by Chevalley Claude;

Author:Chevalley, Claude;
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2018-06-11T04:00:00+00:00

§I. DEFINITION OF THE NOTION OF ANALYTIC GROUP. EXAMPLES

Definition 1. An analytic group is a pair (, G) formed by a manifold, and a group G which satisfy the following conditions: 1) the set of points of coincides with the set of elements of G; 2) the mapping (σ, ) → σ–1 of the manifold × into is everywhere analytic.

The manifold is called the underlying manifold of the analytic group. The underlying topological space of is also called the underlying space of the group. Since every analytic mapping is continuous, the pair formed by the underlying space and the group G is a topological group, which is called the underlying topological group of the analytic group. This topological group is obviously connected and locally simply connected.

The additive group of Rn, associated with the manifold Rn which was defined in Chapter III, §II, p. 73 is an analytic group which we shall again denote by Rn.

We now consider the group GL(n, C). If σ = (xij(σ)) is a matrix in this group, we denote by and the real and imaginary parts of xij(σ). If we assign to σ the point Φ(σ)εR2n2 whose coordinates are the numbers , (arranged in some fixed order), we obtain a homeomorphism, Φ, of GL(n, C) with the subset of R2n2 composed of the points for which

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