1

I am trying to prove the following:

Let $L$ be a Lie algebra and $I$ an Ideal of $L$. There exists a bijection between the Ideals of the quotient algebra $L/I$ and the Ideals of $L$, that contain $I$.

  • Let $J$ be an Ideal of $L$, which contains $I$ $\Rightarrow$ $J/I$ is an Ideal of $L/I$
  • Let $K$ be an Ideal of $L/I$ $\Rightarrow$ $J:=\{z\in L \mid z+I \in K\}$ is an Ideal of $L$, which contains $I$

I already know that this holds true for vector spaces (as it is true for modules). Knowing that, how can I show that this also holds true for Lie algebras?

mathStudent
  • 441
  • Just do it the same way as here, only for Lie algebras. – Dietrich Burde Apr 25 '20 at 19:21
  • My point is that I want to show that the lie bracket is preserved under this bijection. I have already proven that the bijection is true for vector spaces. – mathStudent Apr 25 '20 at 20:14
  • 1
    Have you tried to go through your favorite proof for vector spaces and observe that in your setting all the linear maps that are constructed are homomorphisms of Lie algebras. Then the only additional observation that should be needed is that the preimage of an ideal under a surjective homomorphisms of Lie algebras is again an ideal, which is easily proved directly. – Andreas Cap Apr 26 '20 at 08:50

1 Answers1

1

I think this well-know proof holds for Lie algebras.

Let $J$ be an ideal of $L$ that contain $I$. Consider the map $$\begin{array}\ \phi:& L & \rightarrow & L/I \\ & J &\mapsto & J/I\end{array}$$ Note that $J/I$ is an ideal of $L/I$ since for $X+I \in J/I$ and $Y+I\in L/I$ we have $[X+I,Y+I]=[X,Y]+I$, where $[X,Y]$ is in $J$ because it is an ideal of $L$.

Now, let $K$ be an ideal of $L/I$ and consider the map $$\begin{array}\ \psi:& L/I & \rightarrow & L \\ & K &\mapsto & \{ X\in L\ |\ X+I \in K\}\end{array}$$

We need to check that this set is in fact an ideal of $L$, but this is easy. For $X\in \psi(K)$ and $Y\in L$ we have $[X,Y]\in L$ and $[X,Y] +I=[X+I,Y+I] \in K$, because $K$ is an ideal.

  • $\psi(\phi(J))=\psi(J/I) = \{X\in L\ |\ X+I \in J/I\}=J$.
  • $\phi(\psi(K))=\phi(\{Y\in L\ |\ Y+I \in K\})=\{Y+I\in L/I \ |\ Y+I\in K\} = K$.

This show the bijection.

Júlio César M. Marques
  • 328