From: parendt@newshost.nmt.edu (Paul Arendt)
Subject: Re: Simple group theoretic construction
Date: 7 Dec 1999 14:44:36 -0700
Newsgroups: sci.math
Keywords: Spin groups, local isomorphism of Lie groups
In article <384D0366.BCB19CA7@company.com>,
john Smith wrote:
>It is well known that SU2 is locally isomorphic to SO3.
>How does this extend to higher dimensions?
>I.e. if SU(N) is locally isomorphic to SO(f(N)), what does f look like?
>Is it N^2-1, or someting more complicated?
That's a false premise; the "f" in question doesn't exist. A
slightly re-worded version of your question *is* answerable,
however:
What is the universal covering group of SO(N)?
(The universal covering group shares the Lie algebra.) The answer
is called Spin(N) -- it's an interesting enough question that a
whole set of groups have been defined by the question! Spin(3)
is SU(2), as you noted above. Some others are:
Spin(4) = SU(2) x SU(2)
Spin(5) = Sp(4) (Maybe? I don't remember.)
Spin(6) = SU(4)
SO(2) is just U(1), whose universal covering group is just the real
line under addition. Look under "spin groups" in some group theory
books and you can probably find some others (and check Spin(5) which
I'm not sure of).
>Another, somewhat related, question is:
>Let V and W be a free vectorspaces with a groups G and H (respectively)
>acting on the elements of the respective vectorspaces.
>Let us denote the tensorproduct of V with W as U=VxW
>We know the representation of the products of the groups G and H in the
>tensorspace U is just the Kronecker product.
>Given that the productgroup is SU(N), what are is the possible set of
>candidates for the groups G and H?
Hmmm -- I see two possible questions here, and I'm not sure which one
you're trying to ask. Either you're asking:
- What two groups have SU(N) as a direct product?
or:
- What two groups have representations which tensor product together
to form representations of SU(N)?
The answer to the first is: no interesting ones! You can always have
the trivial one-element group times SU(N) giving SU(N) itself, but
other than that SU(N) is "prime." (There might be certain values
of N for which SU(N) is a direct product, but I don't know of any.
I'm not an expert in this though.)
My answer to the second is: I don't know. Direct *sums* of
representations can sometimes be written as *products* of
representations of the same group, of course.
==============================================================================
From: Robin Chapman
Subject: Re: Simple group theoretic construction
Date: Wed, 08 Dec 1999 08:49:44 GMT
Newsgroups: sci.math
In article <384D0366.BCB19CA7@company.com>,
john Smith wrote:
> It is well known that SU2 is locally isomorphic to SO3.
> How does this extend to higher dimensions?
> I.e. if SU(N) is locally isomorphic to SO(f(N)), what does f look like?
> Is it N^2-1, or someting more complicated?
Lie groups can only be locally isomorphic if their Lie algebras are
isomorphic. In all these examples the Lie algebra will be semisimple
and so the Lie algebras in question can only be isomorphic if their
root systems are identical.
For SU(n) the root systems are of type A_{n-1}. The root systems
of SO(n) are a bit harder to describe.
In general S(2n+1) has root system B_n (n >= 2) and
S(2n) has root system D_n (n >= 4). For small n we get exceptional
cases:
SO(3): root system A_1
SO(4): root system A_1 x A_1
SO(6): root system A_3
So, as you say, SO(3) and SU(2) are locally isomorphic, and it's
well-known that SO(4) is locally isomorphic to SU(2) x SU(2).
The only other case is SO(6) and SU(4). Apart from these there
cannot be any other local isomorphisms between SU(n)s and SO(m)s.
For details see any text on Lie groups.
--
Robin Chapman
http://www.maths.ex.ac.uk/~rjc/rjc.html
"`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'"
Greg Egan, _Teranesia_
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