Update on Sp(4, |R) algebra (like, so much more than you needed to know) --- I did find a 4-subspace: M = [[a,-c,-d,-b],[c,-a,b,d],[d,-a,-b,-c],[b,-d,c,a]] satisfies M' K M = (a^2 + b^2 - c^2 - d^2) K for all real a,b,c,d. Algebra generators x,y,u,v correspond to setting a,b,c,d = 1 individually with other three zero; algebra relations are x x = y y = 1, u u = v v = -1, x y = u v (= K), v y = y v = u x = - x u, u y = y u = x v = -v x. Alas, simple computation now shows that the spinor dimension (of products, reduced modulo scalars) is only 3, against target 10. By analogy with Clifford algebra, the best to be hoped for anyway would be 4_C_2 = 6; achieving 10 almost certainly requires 5 generators. The extra generator cannot contribute to the inner product, for which the skew-symmetric bilinear form is given: so the larger algebra is degenerate. Geometrically, this would correspond to compactification of the space, via attachment of some subspace "at infinity"; the corresponding extra matrix attached to the basis is presumably singular, and annihilates K. [By the way, contrary to a hint dropped earlier, the "spinor norm" (for a vector, the inner product with itself) is distinct from the matrix determinant; though either may be used for normalisation. At least, the norm might be used, were it not for the inconvenient circumstance that in this algebra, the norm of every vector is zero!] I reckon there's a good chance that something useful and interesting is waiting to be unearthed here; and given the notable reluctance of most respectable mathematicians to engage with degenerate algebras, it's quite likely that nobody has yet got around do doing so! Fred Lunnon On 2/27/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
A typographical eagle points out that the earlier
where K = [[1,0],[0,-1]] is the canonical skew-symmetric matrix
should obviously have read
where K = [[0,-1],[1,0]] is the canonical skew-symmetric matrix
Sp(4) of course needs instead K = [[0,0,0,-1],[0,0,-1,0],[0,1,0,0],[1,0,0,0]]; or perhaps K = [[0,-1,0,0],[1,0,0,0],[0,0,0,-1],[0,0,1,0]].
And while I'm at it, I have just realised that
[the] central problem is whether these matrices comprise a vector space
is a gross over-simplification. Rather, it's whether, somewhere within the dimension-11 GSp(4), there exists a dimension-4 vector space, which then generates the whole group via multiplication.
(Sigh ...) WFL