# The Underlying Manifolds of the Special Unitary Lie Groups SU(n)

I want to find the underlying manifolds of Lie Groups $\mbox{SU}(n)$ for general $n$.

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My lecturer told us last year that the only n-spheres that admit a Lie group structure are $\mathbb{S}^1$ and $\mathbb{S}^3$. Moreover I see several question describing why this is so, such as here and here, and understand the arguments therein.

However each Lie group must have some underlying manifold by definition. However I do not know how to find what the manifold is for any given Lie group.

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My motivation for this question is to understand the Lie algebra relation $$\mathfrak{su}(4) \simeq \mathfrak{so}(6)$$ and as such there is some relation between the Lie groups $\mbox{SU}(4)$ and $\mbox{SO}(6)$, with my neive guess being

$$\mbox{SU}(4) / \mathbb{Z}_2 \simeq \mbox{SO}(6)$$

I don't have any very good reason for this however, but it feels right. I note that

$$\mbox{dim}[\mbox{SU}(4)] = 15 \quad \mbox{ and } \quad \mbox{dim}[\mbox{SO}(6)] = 15$$

so we don't immediately think that no such relation should exist. Moreover, I'm encouraged by the relation $\mbox{SU}(2) / \mathbb{Z}_2 \simeq \mbox{SO}(3)$, which I understand and can calculate.

Explicitly, one first notes that both $\mathfrak{su}(2)$ and $\mathfrak{so}(3)$ have the same Lie algebra, namely

$$[T_i, T_j ] = \varepsilon^{ijk} T_k$$

Then one shows that $\mbox{SU}(2)$ has the 3-sphere $\mathbb{S}^3$ as underlying manifold, since any $U \in \mbox{SU}(2)$ can be written like

$$U = a_0 I_2 + i \vec{a} \cdot \vec{\sigma}$$

with

$$a_0^2 + \vec{a} \cdot \vec{a} = 1$$

and this is the equation for the unit 3-sphere $\mathbb{S}^3$, while it can be shown that $\mbox{SO}(3)$ has underlying manifold $\mathbb{RP}^3$, and it remains to note that $\mathbb{S}^3$ is a double cover of $\mathbb{RP}^3$, which motivates the quotient by $\mathbb{Z}_2$, leading to the result.

Now, I have tried thinking about using this approach in my case of $\mbox{SU}(4)$ and $\mbox{SO}(6)$, but I make no progress.

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Perhaps a related point, we can show that the action of $\mbox{SU}(2)$ on $\mathbb{C}^2 \simeq \mathbb{R}^4$ has as orbits 3-spheres $\mathbb{S}^3$ by considering the fact that that the action of $\mbox{SU}(2)$ on $\mathbb{C}^2$ preserves the Hilbert norm, giving a constraint

$$||\vec{z}||^2 = |z_1|^2 + |z_2|^2 = x_1^2 +y_1^2 + x_2^2 +y_2^2$$

for $\vec{z} = (z_1,z_2) = (x_1 + i y_1, x_2 + i y_2) \in \mathbb{C}^2$.

Now this I have had success (I hope!) in generalising, and I find that the action of $\mbox{SU}(n)$ on $\mathbb{C}^n \simeq \mathbb{R}^{2n}$ should have as orbits (2n-1)-spheres $\mathbb{S}^{2n-1} \subset \mathbb{R}^{2n}$. I am unsure if this actually helps me with my original question however.

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