| Summary: | 碩士 === 淡江大學 === 數學研究所 === 65 === In my personal view point, to know how to prove and compute the following familiar results in the theory of Lie groups is very important for us:
1. The general linear group GL(n,k) is open in the manifold (differentiable) M(n,k) (where K=R.C.H.)
2. The special linear group SL(n,k) is closed in GL(n,k). (where R,C.)
3. The classical groups O(n), U(n), Sp(n), SO(n), SU(n), are compact.
4. The Classical groups GL(n,C), GL(N,H), SL(n,R), SL(n,C), U(n), Sp(n), SO(n), SU(n) are pathwise connected.
5. The classical groups SU(n), Sp(n), GL(n,H), SL(n,C) are simply connectted.
6. The group Spin(n) (universal covering group of SO(n)) is a athwise connected, simply connected compact group.
7. The projective space KPn-1 is a pathwise connnectetd compact 2-space. (where K=R,C,H,)
However, all of the facts listing above shall be given in part 1 of this lecture notes. On the other hand, I shall also give an appendix of the computations about some important homogeneous space which have slayed an important role concerning some topics on the topology and meometry of manifolds recently.
In part II, this notes shall construct the Lie groups of type G2 explicitely. Althogh it is will-known for us that the exceptional Lie lgebra of type G2 have only different two Lie groups up to local of autoomorphism of Lie groups, we shall see that two Lie groups can be given the automorphism groups of the Cayles algebra and the split Caylez gebra, respectively. Besides we shall/also check some important top-ogical properties of them. Wish to express
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