# Maximal subgroups of semisimple Lie groups

The problem of finding and classifying the maximal subgroups of simple Lie groups like $SU(3)$ is well known and solved in the literature. What about maximal subgroups of semisimple groups like $SU(3) \otimes SU(3)$?

Are all maximal subgroups of $SU(3) \otimes SU(3)$ trivially obtained from the maximal subgroups of $SU(3)$?

• They have been classified by E. B. Dynkin: Maximal subgroups of semisimple Lie groups and the classification of primitive groups of transformations, Dokl. Akad. Nauk SSSR (1950). – Dietrich Burde Apr 12 '13 at 13:09
• Thanks! Apparently the article has not been translated though so if it's not too much asking, do you know a reference in english? I'm trying to get the original translated in any case but an english reference would be of great help. – user22139 Apr 12 '13 at 15:30
• Fortunately there are english translations, see mathoverflow.net/questions/60315/… – Dietrich Burde Apr 12 '13 at 17:16
• To add to what I wrote in the MO entry linked by D. Burde, the older AMS translation volume has gone out of print by now while the newer AMS collection includes the same two papers by Dynkin (and more). I assume the translations are the same. The paper on maximal subgroups was translated by the well-known group theorist Kurt Hirsch, for example. Since the early papers by Dynkin are lengthy, the early translations can probably be trusted (if you can locate them). By the way, your use of the tensor product symbol (maybe common in physics) is disorienting. Direct product? – Jim Humphreys Apr 13 '13 at 22:55

Maximal (closed) subgroups of semisimple Lie groups have been first classified by E.B. Dynkin (see the reference above). There is indeed a way to "reduce" the problem to the case of a simple Lie group, but it is more complicated than just to take the direct product of the maximal subgroups of the simple factors.

In the example $G=SU(n) \times SU(n)$, or in the example $SU(n)\times_{\mathbb{Z}_n} SU(n)=SU(n)\otimes SU(n)$, we would indeed start with the maximal subgroups of $SU(n)$, which are given by:

(a) $SO(n)$

(b) $Sp(m), m=2n$

(c) $S(U(k) \times U(n-k))=\lbrace (A,B)\in U(k)\times U(n-k) \mid \det(A)\det(B)=1\rbrace$ for $1\le k\le n-1$.

(d) $SU(p)\times_{\mathbb{Z}_d} SU(q)$, where $d=gcd(p,q)$, $pq=n$, $p,q\ge 3$

(e) $\rho(H)$, $H$ simple, $\rho \in Irr_{\mathbb{C}}$ of degree $n$.

Then we should proceed as in Theorem 5.9, 5.10 etc. of the following (modern) reference:

Fernando Antoneli, Michael Forger and Paola Gaviria, Maximal Subgroups of Compact Lie Groups.

I am a bit too lazy to write out the result. Anyway, you don't have to read Dynkin (although this is interesting !).