Maximal compact subgroups
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From: Louis Burnside on 10 Aug 2010 20:22 Hello everybody! A maximal compact subgroup does not exist even for a general Lie group. However, there is the famous Malcev-Iwasawa theorem (e.g. http://books.google.com/books?id=3_BPupMDRr8C&pg=PA263&dq=malcev-iwasawa+theorem&hl=en&ei=ldhhTOSqKYWglAfP0fjuDA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDcQ6AEwAw#v=onepage&q=malcev-iwasawa%20theorem&f=false ). I want to understand, for instance, the simple case when G is compact and totally disconnected: Is the existence of a maximal compact subgroup obvious in this case? Is there a simple proof? Best wishes, Louis.
From: Rupert on 10 Aug 2010 20:47 On Aug 11, 10:22 am, Louis Burnside <burnside.lo...(a)gmail.com> wrote: > Hello everybody! > > A maximal compact subgroup does not exist even for a general Lie > group. > However, there is the famous Malcev-Iwasawa theorem > (e.g. > > http://books.google.com/books?id=3_BPupMDRr8C&pg=PA263&dq=malcev-iwas... > > ). > I want to understand, for instance, the simple case when G is compact > and totally disconnected: > Is the existence of a maximal compact subgroup obvious in this case? > Is there a simple proof? > > Best wishes, > Louis. Do you perhaps mean "locally compact"?
From: Louis Burnside on 10 Aug 2010 21:19 On 10 Ago, 21:47, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Aug 11, 10:22 am, Louis Burnside <burnside.lo...(a)gmail.com> wrote: > > > > > > > Hello everybody! > > > A maximal compact subgroup does not exist even for a general Lie > > group. > > However, there is the famous Malcev-Iwasawa theorem > > (e.g. > > >http://books.google.com/books?id=3_BPupMDRr8C&pg=PA263&dq=malcev-iwas... > > > ). > > I want to understand, for instance, the simple case when G is compact > > and totally disconnected: > > Is the existence of a maximal compact subgroup obvious in this case? > > Is there a simple proof? > > > Best wishes, > > Louis. > > Do you perhaps mean "locally compact"? Hi Rupert! I really mean compact. I just learned that a compact and totally disconnected group is also called a profinite group. So maybe we can somehow build the maximal compact subgroup from the maximal subgroups of the finite quotients of it. I don't know... Best, Louis.
From: David C. Ullrich on 11 Aug 2010 07:13 On Tue, 10 Aug 2010 18:19:25 -0700 (PDT), Louis Burnside <burnside.louis(a)gmail.com> wrote: >On 10 Ago, 21:47, Rupert <rupertmccal...(a)yahoo.com> wrote: >> On Aug 11, 10:22�am, Louis Burnside <burnside.lo...(a)gmail.com> wrote: >> >> >> >> >> >> > Hello everybody! >> >> > A maximal compact subgroup does not exist even for a general Lie >> > group. >> > However, there is the famous Malcev-Iwasawa theorem >> > (e.g. >> >> >http://books.google.com/books?id=3_BPupMDRr8C&pg=PA263&dq=malcev-iwas... >> >> > ). >> > I want to understand, for instance, the simple case when G is compact >> > and totally disconnected: >> > Is the existence of a maximal compact subgroup obvious in this case? >> > Is there a simple proof? >> >> > Best wishes, >> > Louis. >> >> Do you perhaps mean "locally compact"? > >Hi Rupert! > >I really mean compact. Fine. If G is compact then G is a maximal compact subgroup of G. >I just learned that a compact and totally >disconnected group is also called a profinite group. >So maybe we can somehow build the maximal compact subgroup from the >maximal subgroups of the finite quotients of it. >I don't know... > >Best, >Louis.
From: Louis Burnside on 11 Aug 2010 08:59
On 11 Ago, 08:13, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > On Tue, 10 Aug 2010 18:19:25 -0700 (PDT), Louis Burnside > > > > > > <burnside.lo...(a)gmail.com> wrote: > >On 10 Ago, 21:47, Rupert <rupertmccal...(a)yahoo.com> wrote: > >> On Aug 11, 10:22 am, Louis Burnside <burnside.lo...(a)gmail.com> wrote: > > >> > Hello everybody! > > >> > A maximal compact subgroup does not exist even for a general Lie > >> > group. > >> > However, there is the famous Malcev-Iwasawa theorem > >> > (e.g. > > >> >http://books.google.com/books?id=3_BPupMDRr8C&pg=PA263&dq=malcev-iwas... > > >> > ). > >> > I want to understand, for instance, the simple case when G is compact > >> > and totally disconnected: > >> > Is the existence of a maximal compact subgroup obvious in this case? > >> > Is there a simple proof? > > >> > Best wishes, > >> > Louis. > > >> Do you perhaps mean "locally compact"? > > >Hi Rupert! > > >I really mean compact. > > Fine. If G is compact then G is a maximal compact subgroup of G. > > > > >I just learned that a compact and totally > >disconnected group is also called a profinite group. > >So maybe we can somehow build the maximal compact subgroup from the > >maximal subgroups of the finite quotients of it. > >I don't know... > > >Best, > >Louis. What I really meant is maximal compact proper subgroup of G - sorry. Equivalently, maximal closed proper subgroup if G is compact (and Hausdorff). Best, Louis. |