Breaking down SU(N) representation into smaller groups

In summary, the speaker has a shallow understanding of group theory and is trying to generalize a problem involving SU(N) symmetry. They are considering using SU(2) matrices as a basis to simplify their calculations, but are unsure of how to break the SU(4) matrices into sets of SU(2) matrices. The group has no natural way to do so, but the speaker eventually settled on using a combination of SU(n1) and SU(n2) matrices to solve the problem.
  • #1
diegzumillo
173
18
Hi all

I have a shallow understanding of group theory but until now it was sufficient. I'm trying to generalize a problem, it's a Lagrangian with SU(N) symmetry but I changed some basic quantity that makes calculations hard by using a general SU(N) representation basis. Hopefully the details of the problem are not important though, as I just want to rewrite these matrices in a more useful way. Say I have SU(4), it has 15 generators, right, so it looks plausible to replace this basis with 5 sets of SU(2) matrices instead. That would simplify my calculations! But looking at the actual matrices of SU(4) it's hard to see how to break them into SU(2) sets.

This seems to me like basic stuff I don't know. Anyone cares to nudge me in the right direction?
 
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  • #2
diegzumillo said:
Hi all

I have a shallow understanding of group theory but until now it was sufficient. I'm trying to generalize a problem, it's a Lagrangian with SU(N) symmetry but I changed some basic quantity that makes calculations hard by using a general SU(N) representation basis. Hopefully the details of the problem are not important though, as I just want to rewrite these matrices in a more useful way. Say I have SU(4), it has 15 generators, right, so it looks plausible to replace this basis with 5 sets of SU(2) matrices instead. That would simplify my calculations! But looking at the actual matrices of SU(4) it's hard to see how to break them into SU(2) sets.

This seems to me like basic stuff I don't know. Anyone cares to nudge me in the right direction?
Of course it all depends on what you want to achieve, but there is no natural such way. ##\mathfrak{su}(n)## are simple Lie algebras, and representations can only be split by a decomposition into ideals, diagonal matrices if the product is direct, upper triangular matrices if the product is semidirect. As ##\mathfrak{su}(n)## has no ideals and the group only a central discrete group of unities, there is no natural way to do so. You can only change the basis and therewith the matrix representations of the generators. However, the representations of the groups / Lie algebras are all known, at least in principle, which reduces your choices further.

In short: If you restrict yourself to embedded generators of another group, the representations of the larger group will lead you out of them, i.e. the matrices are still full, i.e. no zero blocks.
 
  • #3
Here are the two ways that you can split the 15 generators of SU(4) into 5 groups of SU(2) generators:
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).

SU(4)_1.jpg


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

\begin{equation}SO(4)~\cong~S_i^3~\times~S_j^3\end{equation}

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.

SU(4)_3.PNG



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:

SU(4)_2.PNG
 

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  • #4
Thanks! I appreciate the help. This is an old topic, at the end I went with a very anticlimactic SU(n1)xSU(n2)x some extra matrices (with n=n1+n2). It made sense in the problem at hand.
 

1. What is SU(N) representation?

SU(N) representation refers to the mathematical representation of a special unitary group of matrices with a specific dimension N. It is commonly used in the study of quantum mechanics and particle physics to describe the symmetries of physical systems.

2. Why is it important to break down SU(N) representation into smaller groups?

Breaking down SU(N) representation into smaller groups allows for a simpler and more manageable way of studying and understanding the symmetries of a system. It also helps in identifying patterns and relationships between different subgroups, which can provide insights into the underlying physics of the system.

3. How is SU(N) representation broken down into smaller groups?

SU(N) representation can be broken down into smaller groups through a process called reduction. This involves finding a subgroup of SU(N) that shares the same symmetries as the original group, but with a lower dimension. This can be achieved through mathematical techniques such as group theory and Lie algebras.

4. What are the applications of breaking down SU(N) representation?

The breaking down of SU(N) representation has various applications in physics, particularly in the study of fundamental particles and their interactions. It is also used in other fields such as condensed matter physics, where it helps in understanding the symmetries of materials and their properties.

5. Are there any limitations to breaking down SU(N) representation into smaller groups?

While breaking down SU(N) representation into smaller groups can be useful, it is not always possible for all systems. Some systems may not have any subgroups with lower dimensions, making it impossible to reduce the representation. Additionally, the process of reduction can sometimes be mathematically complex and may require advanced techniques.

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