Dimensional coincidences between Lie groups

Question

There are several exceptional isomorphisms between classical Lie groups that occur in low dimensions; of course it is necessary for the dimensions of the groups to coincide for this to happen! I'm curious if coincidences with pairs of higher dimensional Lie groups having the same dimension have meaning of some sort too, even if it isn't from a full-blown isomorphism. I'll relate some examples of what I'm looking for below.

The dimensions of $SU(n)$ and $SO(m)$ are $n^2-1$ and $m(m-1)/2$, respectively. Small cases where these are equal include $n=2, m=3$ and $n=4,m=6$, which relate to the exceptional isomorphisms $SU(2) \cong Spin(3)$ and $SU(4) \cong Spin(6)$; these and other low dimensional isomorphisms (which can be found here) make it natural to be curious if anything similar occurs in higher dimensions; the Diophantine equations involved are cute (involving a square being one more than a triangular number), which would make positive results in this vein all the more satisfying.

There are infinitely many pairs $(n,m)$ where these dimensions are equal; for example, take $n=11, m=16$. This might make one hope there is a something interesting happening with $SU(11)$ and $SO(16)$. There isn't an exceptional isomorphism with $Spin(16)$ (it's not on the lists of exceptional isomorphisms I've seen, which I assume are complete), but could there still be some relationship between these groups, or some interesting (possibly geometric) consequence to these? Of course, that value of $(n,m)$ is just an example, I'm interested in this phenomenon in general; any Lie groups are fair game.

(as one last example, a fun Diophantine equation comes from asking when a square number can be a triangular number: an example is $dim(U(6)) = 36 = 6^2 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = dim(SO(10))$; it would be very nice if this "meant" something)

Answer 1

There are no special isomorphisms among different simple Lie groups of dimension $> 15$. Still, there are a few things that can be said:

• Different real forms of the same complex simple Lie group have, of course, the same dimension. For example, for any $n > 1$, $\dim SL(n, \Bbb R) = SU(p, n - p)$ for any $p \in \{0, \ldots, n\}$ (both groups are real forms of the complex simple Lie group $SL(n, \Bbb C)$.)
• If you're familiar with the $X_n$ nomenclature for complex simple Lie groups, we have $$\dim_{\Bbb C} B_n = \dim_{\Bbb C} C_n = n (2 n + 1) = n (2 n + 1),$$ that is, $$\dim_{\Bbb C} SO(2 n + 1, \Bbb C) = \dim_{\Bbb C} Sp(2 n, \Bbb C) .$$ One consequence of this coincidence for real groups is that $$\dim SO(2 n + 1) = \dim PSp(n) = n (2 n + 1)$$ for all $n \geq 2$. In both cases the groups are isomorphic only for $n = 2$.

I don't know of anything that can be said about the relationship between generic pairs $G, G'$ of simple Lie groups whose dimensions coincide.