# Relationship Between Symplectic Group and Orthogonal Group

Bacle
Hi, All:

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?

Simon_Tyler
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)
http://www.cns.gatech.edu/GroupTheory/

The Z/2 thing you mentioned is discussed on wikipedia
http://en.wikipedia.org/wiki/Orthogonal_group#Orthogonal_groups_of_characteristic_2

Finally, there might also be some sort of approach through generalized complex geometry
http://en.wikipedia.org/wiki/Generalized_complex_structure
(or maybe not...)

Bacle
Thanks, Simon.

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
formally-defined.

My opinion, in case anyone is interested.

Thanks again, Simon.