$\DeclareMathOperator\GL{GL}$I'm reading the proof of Serre's open image theorem from his book "Abelian $\ell$-adic representations and elliptic curves". This is chapter IV Section 2.2 of the book. Let's assume that $E$ is an elliptic curve without CM over a number field $K$ and $\rho:G_K \rightarrow \GL(V_\ell)$ the associated Galois representation on $ V_\ell = T_\ell \otimes \mathbb{Q} $ where $ T_\ell $ is the Tate module. Then he proves that the image of the Galois representation is open in $\GL(V_\ell)$ w.r.t the $\ell$-adic topology. The only part that I don't understand from the proof is in the very beginning when he deduces that the ($\ell$-adic) Lie algebra of the image contains $\mathfrak{sl}_2$ from the fact that it's centralizer is $\mathbb{Q}_\ell$. Is he using some sort of classification for the Lie subalgebras here?

My main question is this: To me this seems like a Lie algebra analogue of the double centralizer theorem for simple subalgebras of central simple algebras. I was wondering if such a result exists, namely if from the centralizer of the Lie subalgebra being small we can deduce that the subalgebra itself is large in some sense. I'm not sure what should be the conditions and what's the right formulation but I fell like $\GL_2$ being reductive and $\operatorname{SL}_2$ being semi-simple might play a role here.

$\DeclareMathOperator\GL{GL}$I'm reading the proof of Serre's open image theorem from his book "Abelian $\ell$-adic representations and elliptic curves". This is chapter IV Section 2.2 of the book. Let's assume that $E$ is an elliptic curve without CM over a number field $K$ and $\rho:G_K \rightarrow \GL(V_\ell)$ the associated Galois representation on $ V_\ell = T_\ell \otimes \mathbb{Q} $ where $ T_\ell $ is the Tate module. Then he proves that the image of the Galois representation is open in $\GL(V_\ell)$ w.r.t the $\ell$-adic topology. The only part that I don't understand from the proof is in the very beginning when he deduces that the ($\ell$-adic) Lie algebra of the image contains $\mathfrak{sl}_2$ from the fact that it's centralizer is $\mathbb{Q}_\ell$. Is he using some sort of classification for the Lie subalgebras here?

My main question is this: To me this seems like a Lie algebra analogue of the double centralizer theorem for simple subalgebras of central simple algebras. I was wondering if such a result exists, namely if from the centralizer of the Lie subalgebra being small we can deduce that the subalgebra itself is large in some sense. I'm not sure what should be the conditions and what's the right formulation but I feel like $\GL_2$ being reductive and $\operatorname{SL}_2$ being semi-simple might play a role here.

I'm reading the proof of Serre's open image theorem from his book Abelian $\ell$-adic Representations and Elliptic Curves. This is chapter IV section 2.2 of the book. Let's assume that $E$ is an elliptic curve without CM over a number field $K$ and $\rho:G_K \rightarrow GL(V_\ell)$ the associated Galois representation on $ V_\ell = T_\ell \otimes \mathbb{Q} $ where $ T_\ell $ is the Tate module. Then he proves that the image of the Galois representation is open in $GL(V_\ell)$ w.r.t the $\ell$-adic topology. The only part that I don't understand from the proof is in the very beginning when he deduces that the ($\ell$-adic) Lie algebra of the image contains $\mathfrak{sl}_2$ from the fact that it's centralizer is $\mathbb{Q}_l$. Is he using some sort of classification for the Lie subalgebras here?

My main question is this: To me this seems like a Lie algebra analogue of the double centralizer theorem for simple subalgebras of central simple algebras. I was wondering if such a result exists, namely if from the centralizer of the Lie subalgebra being small we can deduce that the subalgebra itself is large in some sense. I'm not sure what should be the conditions and what's the right formulation but I fell like $GL_2$ being reductive and $SL_2$ being semi-simple might play a role here.

$\DeclareMathOperator\GL{GL}$I'm reading the proof of Serre's open image theorem from his book "Abelian $\ell$-adic representations and elliptic curves". This is chapter IV Section 2.2 of the book. Let's assume that $E$ is an elliptic curve without CM over a number field $K$ and $\rho:G_K \rightarrow \GL(V_\ell)$ the associated Galois representation on $ V_\ell = T_\ell \otimes \mathbb{Q} $ where $ T_\ell $ is the Tate module. Then he proves that the image of the Galois representation is open in $\GL(V_\ell)$ w.r.t the $\ell$-adic topology. The only part that I don't understand from the proof is in the very beginning when he deduces that the ($\ell$-adic) Lie algebra of the image contains $\mathfrak{sl}_2$ from the fact that it's centralizer is $\mathbb{Q}_\ell$. Is he using some sort of classification for the Lie subalgebras here?

My main question is this: To me this seems like a Lie algebra analogue of the double centralizer theorem for simple subalgebras of central simple algebras. I was wondering if such a result exists, namely if from the centralizer of the Lie subalgebra being small we can deduce that the subalgebra itself is large in some sense. I'm not sure what should be the conditions and what's the right formulation but I fell like $\GL_2$ being reductive and $\operatorname{SL}_2$ being semi-simple might play a role here.