From: parendt@nmt.edu (Paul Arendt)
Newsgroups: sci.math.research
Subject: Re: Q: generators of symplectic group
Date: 16 Dec 1998 00:30:03 -0600
john baez wrote:
>
>I thought Giedke was talking about "infinitesimal generators", and was
>asking for an explicit basis of the Lie algebra of the symplectic group.
>Physicists often use the term "generators" in this sense. And of course
>some very nice explicit bases of the Lie algebra of symplectic group are
>known. In the case n = 1 there's a nice basis like this:
>
>1 0
>0 -1
>
>0 1
>0 0
>
>0 0
>1 0
>
>Something similar should work in higher dimensions...
It's not difficult to work out. Writing the symplectic form J
(of a 2n-real-dimensional vector space) in the standard fashion
J = 0 I
-I 0
and considering a curve (through the identity at t=0) of
symplectic transformations S(t) which satisfy (by definition)
S(t) J S(t)^T = J ( ^T denotes transpose )
we can differentiate at t=0 to get a defining equation for the Lie
algebra:
L J + J L^T = 0
A general basis (*) of the Lie algebra then looks like (in n by n
block matrix notation):
A 0 where A is an arbitrary n by n matrix
0 -A^T
0 B where B is symmetric
0 0
0 0 where C is symmetric.
C 0
There are therefore n^2 + 2 ( n(n+1)/2 ) = 2 n^2 + n independent
generators of the symplectic group.
--------------------------
(*) A "basis" would really consist of giving a full basis for each
type of matrix, but that's way too much typing.