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What about mentioning Killing and Cartan's classification of all possible compact simple Lie algebras? (i.e. the classical groups SU(N), SO(N), and Sp(N), and the exceptional algebras G2, F4, E6, E7, and E8.) -- CYD
Yup, that's important. Maybe it fits better on Lie algebra though? AxelBoldt, Monday, May 27, 2002
This statement mentioned in the article: "If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra g over F there is a unique (up to isomorphism) simply connected Lie group G with g as Lie algebra." is this really true? shouldn't we also require that the Lie group be compact and semisimple, for this to hold? like for example, don't Spin(n) and Spin(p,q) have the same Lie algebra (where p+q=n)? only Spin(n) is compact and semisimple, and the other isn't.Lethe 05:05, 9 Mar 2004 (UTC)
- Yes, I think it is true. Spin(p+q) and Spin(p,q) have different real Lie algebras. However, if you complexify you obtain isomorphic complex Lie algebras which tells you that Spin(p+q;C) and Spin(p,q;C) are isomorphic complex Lie groups. -- Fropuff 05:37, 2004 Mar 9 (UTC)
That's correct. In broad terms complexifying the Lie algebra loses information about the group, but simplifies the algebra.
Charles Matthews 10:23, 9 Mar 2004 (UTC)
The article defines Lie group in the category of analytic maps. That's surely the wrong definition though - must be the smooth category, even if analytic can be proved a posteriori.
Charles Matthews 18:29, 7 Apr 2004 (UTC)
I think that the Lorentz group ought to be listed among the examples, because of its significance in physics: it is the point symmetry group of Minkowski spacetime. It is the group of 4x4 matrices L with the property L^T g L = g where g is the diagonal matrix with diagonal elements -1, 1, 1, 1 (or -1, 1, 1, 1 if you prefer that sign convention). It is six dimensional: three of the generators are infinitesimal rotations, and the other three are infinitesimal 'boosts' (changes in the velocity of the observer). I'm afraid I don't know the proper mathematical language.
-- Alistair Turnbull 8 Apr 2004
The Lorentzgroup is a special case of the pseudo-orthogonal Liegroups, defined as the automorphisms of a pseudo-orthogonal vector space. If +..+-....- is the signature of the symmetric bilinear form, these groups O(+...+-....-,R) are among the semisimple Liegroups, which were classified by Cartan. Likewise the pseudo-unitary and symplectic groups should be included as well. Hannes Tilgner