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.