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I am following Brian Hall's "Lie groups, Lie algebra and representations". Compact Symplectic groups (Sp(2n;R)) are intersection of skew-symmetric bilinear forms preserving Symplectic groups and inner product preserving unitary groups. I am having problem visualizing this group. my question is, why this group is important? what are the examples like for n>1?

Hemanta Mandal
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  • IMHO Groups (and its subgroups) that preserve some fixed structure (such as a Riemannian metric, inner product or complex structure or ) are always useful and interesting for recognizing the underlying Riemannian manifold. – C.F.G Sep 15 '21 at 06:22
  • @C.F.G that somehow answers parts of my question. can you provide an possible use of this group? – Hemanta Mandal Sep 15 '21 at 06:27
  • It's hard to understand what you would consider an appropriate answer to this question. What do you consider the uses of $SO(n)$ or $U(n)$ to be? – Michael Albanese Sep 15 '21 at 10:35
  • @MichaelAlbanese GL(n) acts as group of invertible linear transformations while SO(2) arise as group of rotation in the plane $R^{2}$, O(1,1) preserve the hyperbolas and so on...so all I am looking for is a geometrical description of the group or kind of defining property. – Hemanta Mandal Sep 15 '21 at 15:02

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