# Lie subgroups of SU(4)

Other than subgroups of SU(3), what are the Lie subgroups of SU(4)? Assume that the subgroup is closed but not necessarily connected.

Any references would be much appreciated.

• you maybe mean $S(U(3)\times U(1))$ (which is one dimension bigger than $SU(3)$). Probably you want to ask about irreducible subgroups (= irreducible on $\mathbf{C}^4$). Maybe you want connected, or are you asking about finite subgroups as well. – YCor Jan 10 '13 at 0:06
• you could take $H=S(U(2)\times U(2))$, where each $U(2)$ acts on $(\mathbb C)^2$ the standard representation, and the representation of $H$ is the tensor product of these standard representations. This is also $SO(4)\subset SU(4)$. This is not contained in $SU(3)$. You could take finite subgroups of $H$ as well esentially of the form $A\times A$ where $A$ acts irreducibly on ${\mathbb C}^2$. There is also the maximal compact of $Sp_4({\mathbb C})\subset SL_4({\mathbb C})$ which acts irreducibly on ${\mathbb C}^4$. – Venkataramana Jan 10 '13 at 0:18
• This question is worded very much like a homework problem. It would be improved if you included some of your background and motivation, and elaborated a bit on what applications you have in mind, so that references can be more directed. Please read mathoverflow.net/howtoask for further recommendations on what makes a good MathOverflow question. – Theo Johnson-Freyd Jan 10 '13 at 0:21
• Maybe this could help: B. Gruber and M. Ramek, “Boson and Fermion Operator Realizations of su(4) and its Semisimple Subalgebras”, in “Symmetries in Science VII”, ed. B. Gruber and T. Otsuka, Plenum Press, New York, 1994. I think they also wrote a computer program that tells you su(4) reps decompose as reps of its subalgebras. – Uwe Franz Jan 10 '13 at 16:21
• Thank you for the comments. To be more specific, this arose in the context of a universality proof for a quantum gate set. I have a finite set $S$ of four-by-four special unitary matrices, and I want to know what group of matrices $G$ is densely generated by taking finite products from $S$. I can already show that $G$ is continuous, and the representation of $G$ as special unitary matrices over $\mathbb{C}^4$ is irreducible. – Adam Bouland Jan 10 '13 at 18:03

• Adam: see, you do this just by iteration. First list the maximal subgroups of $SU_n$. Then, for each one in this list, you repeat the procedure and list all of the maximal subgroups of it and so on. Eventually you will have all the subgroups of $SU_n$ listed. – Claudio Gorodski Feb 19 '13 at 19:38