|Apr20-11, 09:14 PM||#1|
Relationship Between Symplectic Group and Orthogonal Group
Given a simplectic vector space (V,w), i.e., V is an n-dim. Vector Space ( n finite)
and w is a symplectic form, i.e., a bilinear, antisymmetric totally isotropic and
non-degenerate form, the simplectic groupSp(2n) of V is the (sub)group of GL(V) that
preserves this form. Similarly, given a bilinear, symmetric non-degenerate form q in V,
the orthogonal group O(n) is the subgroup of Gl(V) that preserves q.
Question: is there some relationship between these two groups under some conditions
, i.e containment, overlap, etc? I think the two groups agree when we work with Z/2-
coefficients (since 1=-1 implies that symmetry and antisymmetry coincide), but I am
clueless otherwise. I have gone thru Artin's geometric algebra, but I cannot get
a clear answer to the question.
Anyone know, or have a ref?
Thanks in Advance.
|Apr20-11, 11:07 PM||#2|
You can think of the symplectic and orthogonal groups as being related through negative dimensions. This is discussed in Cvitanović's Birdtrack book (and references within)
The Z/2 thing you mentioned is discussed on wikipedia
Finally, there might also be some sort of approach through generalized complex geometry
(or maybe not...)
|Apr23-11, 10:08 AM||#3|
Just for anyone else who may be interested, my opinion of E.Artin's treatment
of orthogonal and symplectic groups is not --by his own admission--an in-depth
treatment. In addition, I found his conversational style difficult to follow; while
a more informal treatment may be somewhat dry, it is nice to have accurate
references, instead of statements like "the property we wanted", which is never
My opinion, in case anyone is interested.
Thanks again, Simon.
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