# Complexification of $\mathbb{C}\mathbb{P}^3$ as a coset

A way to write the CP3 (complex projective space) is by the following coset (space):

$$\mathbb{C}\mathbb{P}^3=\frac{SU(4)\times SU(2)}{U(3)\times SU(2)}$$

My question is, does the complexification of a product coset is simply the product coset of its complexified factors? I mean

$$\mathbb{C}\mathbb{P}^3_{\mathbb{C}}=\frac{SU(4)_{\mathbb{C}}\times SU(2)_{\mathbb{C}}}{U(3)_{\mathbb{C}}\times SU(2)_{\mathbb{C}}}$$

How can I complexify, for example

$$SU(4)\times SU(2)$$

I know that

$$SU(4)_{\mathbb{C}}\equiv SL(4,\mathbb{C})$$

and

$$SU(2)_{\mathbb{C}}\equiv SL(2,\mathbb{C})$$

So

$$\left(SU(4)\times SU(2)\right)_{\mathbb{C}}\equiv SL(4,\mathbb{C})\times SL(2,\mathbb{C})$$

is true? If not, what it is?

• $G/H$ is not a coset, it is a coset space. The elements of $G/H$ are cosets $gH$. I don't know what you're talking about with the Gauss decomposition and the equation $\mathbb{CP}^3_{\mathbb{C}}=\mathbb{CP}^3\oplus\mathfrak{b}$ doesn't mean anything to me. But I expect complexification to commute with products, i.e. $(G\times H)_{\Bbb C}\cong G_{\Bbb C}\times H_{\Bbb C}$. Jul 4, 2019 at 2:57
• @runway44 I corrected the terminology. Jul 4, 2019 at 12:54