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[SOLVED] Pseudo orthogonal group |
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| Oct12-06, 04:29 AM | #154 |
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[SOLVED] Pseudo orthogonal group
Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #155 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #156 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #157 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #158 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #159 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #160 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #161 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #162 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #163 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #164 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #165 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #166 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #167 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #168 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #169 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
| Oct12-06, 04:29 AM | #170 |
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Arnold Neumaier wrote:
> In particular, SO(p,q) is connected if the product pq is even, > and has two connected components otherwise. This is not true. SO(p,q) has always two connected components if pq>0. At least with the standard definition of SO(p,q) that I repeat in another message on this thread. In fact, it is very easy to see that there are always *at least* two connected components; cf. my other message. > The book by > R. Gilmore, > Lie groups, Lie algebras, and some of their applications > Wiley, New York 1974 > contains a lot of material about specific classical groups. > The above result is stated there (without proof) on p. 199. I haven't looked it up. Maybe a clash of notation? -- Marc Nardmann |
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