## [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

 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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