# When can a matrix $\Lambda\in SL(4,R)$ be represented by $SL(2,C)$?

I was just curious if anyone knows what kind of constraints one can place on $$SL(4,R)$$ the set of 4x4 invertible matrices with unit determinant to obtain: $$SL(2,C)$$ the set of 2x2 invertible complex matrices with unit determinant. I'm guessing that if one places a particular constaint on the former set then they are isomorphic to the latter?

Essentially I'm trying to show that a particular set of matrices I have might be represented in terms of $$SL(2,C)$$ rather than the current form I have them in $$SL(4,R)$$. Thanks a ton!

• This would be easier to answer, I think, if you stated what the current form you have them in is – Omnomnomnom Dec 20 '18 at 23:37
• Are you aware of the usual way of representing $GL(n,\Bbb C)$ in $GL(2n,\Bbb R)$? – Omnomnomnom Dec 20 '18 at 23:38
• @Omnomnomnom No, I wasn't, but I think the below answer straightened me out! (: – R. Rankin Dec 21 '18 at 3:24

If $$\Lambda$$ is of the form: $$\Lambda = \begin{bmatrix} a&b&c&d \\ -b&a&-d&c \\ e&f&g&h \\ -f&e&-h&g \end{bmatrix}$$ then we can isomorphically map it to: $$\begin{bmatrix} a+ib&c+id \\ e+if&g+ih \end{bmatrix}$$
• In terms of Kronecker products (tensor products), we can write this association as $$\pmatrix{a&c\\e&g} + i\pmatrix{b&d\\f&h} \leftrightarrow \pmatrix{a&c\\e&g}\otimes\pmatrix{1&0\\0&1} + \pmatrix{b&d\\f&h}\otimes \pmatrix{0&1\\-1&0}$$ – Omnomnomnom Dec 21 '18 at 5:12