Re: [math-fun] Lie group terminology
The example of this kind of thing mentioned most often seems to be SL(2,R)^, which means the unique simply connected Lie group with the same Lie algebra as SL(2,R). Where SL(2,R) is defined as the matrix group SL(2,R) = {n x n real matrices with determinant = 1}. Then the simply connected Lie group SL(2,R)^ is not a subgroup of the group of invertible m x m matrices for any m. —Dan Andy Latto wrote: ----- The prototypical examples of Lie groups are matrix groups, subgroups of GL(n), the group of invertible n x n matrices. I think it's almost, but not quite, true, that all Lie groups are subgroups of matrix groups. (the "almost" is that I think there are a few (families of?) Lie groups that are double-covers of matrix groups, but aren't matrix groups themselves. ... ... -----
If SL(2,R)^ can’t be written as a group of matrices, is it something almost as nice, such as a quotient group of a group of matrices? Jim Propp On Friday, October 12, 2018, Dan Asimov <dasimov@earthlink.net> wrote:
The example of this kind of thing mentioned most often seems to be
SL(2,R)^, which means the unique simply connected Lie group with the same Lie algebra as SL(2,R).
Where SL(2,R) is defined as the matrix group
SL(2,R) = {n x n real matrices with determinant = 1}.
Then the simply connected Lie group SL(2,R)^ is not a subgroup of the group of invertible m x m matrices for any m.
—Dan
Andy Latto wrote:
----- The prototypical examples of Lie groups are matrix groups, subgroups of GL(n), the group of invertible n x n matrices. I think it's almost, but not quite, true, that all Lie groups are subgroups of matrix groups. (the "almost" is that I think there are a few (families of?) Lie groups that are double-covers of matrix groups, but aren't matrix groups themselves. ... ... -----
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