Is this proof that $\mathrm{SU}(2)$ cannot be isomorphic to $\mathrm{SO}(1,3)$ valid?

Question

It seems intuitively obvious to me that there cannot be an isomorphism between $\mathrm{SU}(2)$ and $\mathrm{SU}(2)\times\mathrm{SU}(2)$ where $\mathrm{SU}(2)$ is the Lie Group with the Pauli matrices as generators and $\mathrm{SU}(2)\times\mathrm{SU}(2)$ is the Direct Product whose generators are formed from the generators of $\mathrm{SU}(2)$ by putting them into $2\times2$ matrices in block diagonal form.

(I was told this is a correct method of taking the Direct Product in an answer to a prior question. See the response "that is one method of taking the Direct Product":

https://physics.stackexchange.com/questions/28505/how-do-i-construct-the-su2-representation-of-the-lorentz-group-using-su2)

And, if so, is the following argument valid?

$\mathrm{SU}(2)$ is not isomorphic to $\mathrm{SO}(1,3)$ because $\mathrm{SO}(1,3)$ is isomorphic to $\mathrm{SU}(2)\times \mathrm{SU}(2)$. If $\mathrm{SU}(2)$ is isomorphic to $\mathrm{SO}(1,3)$, then $\mathrm{SU}(2)$ is isomorphic to $\mathrm{SU}(2)\times \mathrm{SU}(2)$. But, that is false, therefore $\mathrm{SU}(2)$ cannot be isomorphic to $\mathrm{SO}(1,3)$.

Answer 1

$SO(3,1)$ is not isomorphic to $SU(2)\times SU(2)$ (notice e.g. that $SU(2)\times SU(2)$ is compact while $SO(3,1)$ is not), but rather to $SL(2,\mathbb{C})/\pm1$. (however $SL(2,\mathbb{C})$ and $SU(2)\times SU(2)$ are two real version of the same complex group $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$.) Anyway, $SO(3,1)$ is not isomorphic to $SU(2)$ simply by dimension count: the first has dimension $6$ while the second has $3$.

Answer 2

barry, perhaps some more comments about the differences between mathematicians and physicists are in order. In my experience, physicists frequently do not distinguish between Lie groups and their Lie algebras, whereas mathematicians are generally careful to make this distinction. There are many pairs of Lie groups which are not isomorphic but which have isomorphic Lie algebras, and for many purposes this is "close enough."

How would you characterize the relationship between SU(2)XSU(2) and the Lorentz Group?

They have isomorphic Lie algebras.