# Reference request: Relation between $Sp(N)$, $Spin(N)$, $SU(N)$ groups and physics

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I want to understand the relationship of the so common $SU(N)$ and $SO(N)$ groups in physics with the symplectic group which I think is the double cover of the first and the Spin groups $Spin(N)$.

Is there a good reference, mainly for physicists.

This post imported from StackExchange Physics at 2015-01-10 13:18 (UTC), posted by SE-user Marion

asked Jan 9, 2015
recategorized Jan 10, 2015
The symplectic group is not a double cover of $SU(N)$. It is the spin group $Spin(N)$ which is a double cover of $SO(N)$.

## 1 Answer

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I will give references to more general topics. The references, of a combination of them do include though all the information you ask for. I think that you can try to read Pierre Ramond's Group Theory; a Physicist's Survey where there is some information on the Weyl groups. Then, a very nice exposition is this book I recently found titled Lie Groups and Lie Algebras; A Physicists Perspective. Some nice information can be found on the blog post by Lubos on Exceptional Lie Groups.

Also, indeed $Spin(N)$ is the universal double cover of $SO(N)$ for $N \geq 3$ otherwise it is not universal.

answered Jan 11, 2015 by (3,535 points)

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