Talk:Symplectic matrix

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[edit] J vs. Ω

On 19 april 2005, User:Fropuff changed the notation from J to Ω to "avoid confusion with complex structure". I rather want to change the notation back; we want to confuse the two, right? Unless we're trying to reserve J for the 2x2 case only, and Ω for the nxn case? linas 23:35, 18 March 2006 (UTC)

Never mind, I just hacked this article to say that the 2x2 matrix is called J. I wanted to link to this article from an article having J in it. linas 23:41, 18 March 2006 (UTC)

No, we don't! These are two very different things and should be distinguished. In particular, one could easily choose a basis for which Ω² ≠ −1, whereas this is an essential quality of a complex structure. Moreover, J should be understood as a linear transformation wheres Ω is a bilinear form. Given a hermitian structure on a vector space, J and Ω are related via

That J and Ω have the same coordinate expression (up to an overall sign) is simply a consequence of the fact that g is usually to be the identity matrix. -- Fropuff 23:57, 18 March 2006 (UTC)

Excellent point. Since the distinction is all too easily missed, I copied this into the article. FYI, the article on Hermitian manifold defines this, but with a lower-case &oemga; instead. (and no minus sign). 67.100.217.178 04:33, 30 March 2006 (UTC)