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  SU(2) subgroups of SU(4)?

+ 3 like - 0 dislike

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 (15 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

2 Answers

+ 4 like - 0 dislike

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 (720 points) [ no revision ]
+ 3 like - 0 dislike

You can split the 15 generators of SU(4) into 5 groups of SU(2) generators, below are two ways to do that. The table used below is a product table, the generators in the bottom right square are the products of the first row with the first column.Each of the 5 groups is an anti-commuting triplet. Each of them can be used as a base of SU(2).

This uses the 6 generators of SO(4) below,  SO(4) has two anti-commuting triples while the triples commute between each other.


The SO(4) matrices are given below. Note that the colored 3x3 sub matrices are the SO(3) rotation matrices.

Both  SO(4) triples are also alternative bases for quaternians.

Where $*$ is complex conjugation. Further, to get the x,y,z correspondence to the $SO(3)$ generators as mentioned above we have re-associated the x,y,z coordinates with the Pauli matrices as follows: $\sigma^x\!=\!\sigma^3,~\sigma^y\!=\!-\sigma^2,~\sigma^z\!=\!\sigma^1$

Note that this representation is much cleaner as some horrible ones that extend SU(3) to SU(4) 

The full table written out gives:


answered Oct 16, 2018 by Hans de Vries (90 points) [ no revision ]

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