Topology on the Classical Groups and the Exceptional Lie Groups of Type G2

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...

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Main Author: 馬建由
Other Authors: 橫田一郎
Format: Others
Language:zh-TW
Online Access:http://ndltd.ncl.edu.tw/handle/37405777565893009998
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spelling ndltd-TW-065TKU034790012016-06-13T04:16:44Z http://ndltd.ncl.edu.tw/handle/37405777565893009998 Topology on the Classical Groups and the Exceptional Lie Groups of Type G2 馬建由 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 橫田一郎 學位論文 ; thesis 0 zh-TW
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description 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
author2 橫田一郎
author_facet 橫田一郎
馬建由
author 馬建由
spellingShingle 馬建由
Topology on the Classical Groups and the Exceptional Lie Groups of Type G2
author_sort 馬建由
title Topology on the Classical Groups and the Exceptional Lie Groups of Type G2
title_short Topology on the Classical Groups and the Exceptional Lie Groups of Type G2
title_full Topology on the Classical Groups and the Exceptional Lie Groups of Type G2
title_fullStr Topology on the Classical Groups and the Exceptional Lie Groups of Type G2
title_full_unstemmed Topology on the Classical Groups and the Exceptional Lie Groups of Type G2
title_sort topology on the classical groups and the exceptional lie groups of type g2
url http://ndltd.ncl.edu.tw/handle/37405777565893009998
work_keys_str_mv AT mǎjiànyóu topologyontheclassicalgroupsandtheexceptionalliegroupsoftypeg2
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