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SU(2) subgroups of SU(4)?

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The Wikipedia article on Gell-Mann matrices states that there are 3 independent SU(2) subgroups of SU(3). One of them, for example, is given by the generators $\{ \lambda_1, \lambda_2, \lambda_3 \}$, which satisfy the commutation relation of the $\mathfrak{su}(2)$ algebra.

How can I found similar subgroups of SU(4) such that their combination satisfy a commutation relation of the form $[t_a, t_b] = \epsilon_{abc} t_c$ as well?

So far I am aware of three such ways - for example the matrices A, B and B, where $B= i( t_2 + t_{14}) $, $C= i(t_5 - t_{12})$ and $D= i (t_7 + t_{10})$ and $t_i$ are the 4x4 generators of SU(4), obey the above commutation relation.

Are there any more independent ways?

This post imported from StackExchange Mathematics at 2014-11-07 11:08 (UTC), posted by SE-user itsqualtime
asked Nov 6, 2014 in Mathematics by itsqualtime (10 points) [ no revision ]
Related: mathoverflow.net/q/118484 mathoverflow.net/a/65530 Do you have a physical motivation for looking at $\mathrm{SU}(4)$? If not, I think this would be better suited at math.SE.

This post imported from StackExchange Mathematics at 2014-11-07 11:08 (UTC), posted by SE-user ACuriousMind
Yes, we are looking at higher dimensional gauge theories (color electrodynamics). I was actually aware of that question and I have checked all the suggested references - unfortunately most of them are a bit too involved and general (in the case of SU(N), I might have to go that route), but I was hoping this question had already been addressed within the context of SU(4) gauge theories.

This post imported from StackExchange Mathematics at 2014-11-07 11:08 (UTC), posted by SE-user itsqualtime
While there are many pretty results about $\mathrm{SU}(N)$ gauge theories, I know not about such subgroup results. If you were searching for the maximal torus or a similarly "special" subgroup, there are methods, I think, but I can't see anything special about $\mathrm{SU}(2) \subset \mathrm{SU}(4)$. Is there?

This post imported from StackExchange Mathematics at 2014-11-07 11:08 (UTC), posted by SE-user ACuriousMind

1 Answer

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The Dynkin diagram of $SU(4)$ has 3 nodes, which means that it carries three elements in it's Cartan subalgebra. Consider those as three possible choices for $J_z$. Each of those can be attached with a pair of raising/lowering operators $J_{\pm}$ -- to create one $SU(2)$ algebra each.

I presume that argument can be translated to a statement about groups, but I don't think I'm equipped to do that.

This post imported from StackExchange Mathematics at 2014-11-07 11:08 (UTC), posted by SE-user Siva
answered Nov 6, 2014 by Siva (710 points) [ no revision ]

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