I am learning something about lattices in algebraic groups. Consider the algebraic group $\mathrm{SL}_2\times \mathrm{SL}_2$. What are the $\mathbb{Q}$-forms of such groups?
$\begingroup$
$\endgroup$
3
-
2$\begingroup$ There are at least these ones: The non-$Q$-simple ones are the $G_1\times G_2$ where $G_1,G_2$ are $Q$-forms of $SL_2$. The $Q$-simple ones (or some of them?) are the Weil restrictions "$G(K)$" where $G$ is a form of $SL_2$ and $K$ is a quadratic extension of $Q$. $\endgroup$– YCorCommented Dec 5, 2014 at 12:04
-
$\begingroup$ The first case is to some extent discussed in the examples section of de.wikipedia.org/wiki/Arithmetische_Gruppe#Beispiele $\endgroup$– ThiKuCommented Dec 5, 2014 at 14:07
-
4$\begingroup$ YCor's list is exhaustive. In general, the (connected and) simply connected semisimple $k$-groups are precisely the direct products of Weil restrictions $\prod_i {\rm{R}}_{k'_i/k}(G'_i)$ for finite separable extensions $k'_i/k$ and an absolutely simple simply connected semisimple $k'_i$-group $G'_i$ for each $i$. Here, $\sum [k'_i:k]$ is the number of simple factors of $G_{k_s}$, so in your case $\sum [k'_i:k]=2$, recovering the options listed by YCor. The idea of the proof is that ${\rm{Gal}}(k_s/k)$ permutes the simple factors of $G_{k_s}$. $\endgroup$– user74230Commented Dec 5, 2014 at 14:30
Add a comment
|