Re: Complexification of representation
- From: Timothy Murphy <tim@xxxxxxxxxxxxxxxxxxxxxx>
- Date: Fri, 21 Oct 2005 12:01:06 +0100
James wrote:
> Ultimately I am trying to show that if I denote the representation as Ad :
> SU(2) ----> GL(su(2)), I am trying to show that Ad (x) C, its
> complexification, is irreducible. This is why I asked the question above.
> Why is Ad (x) C irreducible?
I find your notation slightly odd.
I take it you are asking why the 3-dimensional representation of SU(2)
over C in the vector space C su(2)
(the complexification of the Lie algebra su(2))
is irreducible.
I would have thought the easiest way to see that this is true,
if you don't mind using a little linear group theory,
is that this corresponds to the adjoint representation
ad(X): Z -> [X,Z] of su(2) (or rather its complexification C su(2).
(Every representation of a linear group G over C
gives rise to a representation of its complexified Lie algebra C LG
in the same space; and the representation of G is irreducible if
the representation of C LG is irreducible.)
>From this point of view, the representation of SU(2) is irreducible
because the Lie algebra C su(2) is simple,
since a subspace of C su(2) stable under SU(2)
will correspond to an ideal in the Lie algebra.
An alternative method of proving the result
would be to consider the restriction of the representation of SU(2)
to the subgroup U(1) formed by the diagonal matrices.
--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
.
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