From: "Michael Weiss" <michael@spamfree.net>
Subject: Re: Lie groups for web crawlers
Date: Thu, 19 Oct 2000 08:31:55 -0400
Newsgroups: sci.physics.research
Summary: [missing]

My guess:

> 3) Every finite dimensional Lie group is isomorphic to a matrix group.

John Baez:

> No!  It ought to be true, but it's not: the math gods were in a bad
> mood that day.  Take the universal cover of SL(2,R).  This has no
> faithful finite-dimensional representations, so it's not isomorphic to
> a matrix group.  At times I have known the proof of this, but now I
> don't, so don't ask me why it's true, unless you wish to make me
> think.

According to Andrew Baker's notes on Lie groups:
http://www.maths.gla.ac.uk/~ajb
the Heisenberg groups Heis_n are Lie groups but not matrix groups for all
n>=3.  And he says that the Lie algebra heis_3 is essentially just the
familiar algebra generated by our old friends p and q on the real line:
qf(x) = xf(x), pf(x) = (d/dx)f(x).

And he gives a proof!  (Which I haven't read.)  So I guess you're off the
hook.

> If we restrict attention to *compact* Lie groups, we're okay: they
> are all isomorphic to matrix groups.

That sounds better.  I guess that somehow falls out of the Cartan
classification theorem?