In the article "An algorithm determining the difference Galois group of second order linear difference equations" by Peter Hendriks from 1998, Lemma 4.1 is a consequence of, among other things, ``the well-known classification of the algebraic subgroups of $GL(2,\overline{\mathbb{Q}})$''. What's that about?
If it were well-known, then I should be able to Google the result, or it would show up in a graduate-level seminar on linear algebraic groups. After hours of searching I haven't found it, and Hendriks hasn't cited anything, so I'm lost. Do people have ideas about what he's referring to?
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If it's helpful, I can include the statement of Lemma 4.1 in case it looks like something that people have seen before.
Lemma 4.1: The algebraic subgroups $G$ of $GL(2,\overline{\mathbb{Q}})$ that can occur as difference Galois groups are:
- Any reducible subgroup of $GL(2,\overline{\mathbb{Q}})$ with $G/G^0$ finite cyclic.
- Any infinite imprimitive subgroup of $GL(2,\overline{\mathbb{Q}})$ with $G/G^0$ finite cyclic.
- Any algebraic subgroup containing $SL(2,\overline{\mathbb{Q}})$.
For context, $G$ is the difference Galois group associated to the system of difference equations $\phi y = Ay$, where $A$ is a $2\times 2$ matrix with entries in $\overline{\mathbb{Q}}(z)$, the map $\phi$ is the $\overline{\mathbb{Q}}$-linear automorphism of $\overline{\mathbb{Q}}(z)$ given by $\phi(z)=z+1$ (the ``shift automorphism''), and $y$ is a $2\times 1$ vector to be solved for. After creating a Picard-Vessiot extension $R$ of $\overline{\mathbb{Q}}(z)$ we can define the difference Galois group $G$ to be the difference $\overline{\mathbb{Q}}$-automorphisms of $R$. Turns out this is a linear algebraic group, and $G_0$ is the identity component of this group. Fun fact, $G/G^0$ is necessarily finite cyclic, which is why this condition appears in items 1 and 2 of Lemma 4.1.
I suspect the classification of the algebraic subgroups of $GL(2,\overline{\mathbb{Q}})$ realize themselves as the reducible, infinite imprimitive, and contains $SL(2,\overline{\mathbb{Q}})$ cases in Lemma 4.1. I still haven't been able to use this to find the classification online.