2
$\begingroup$

Engel's Theorem states that: Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V \neq 0$, then there exists nonzero $v \in V$ for which $L.v = 0$.

If $\dim V = n$, a flag in $V$ is a chain of subspaces $0 = V_{0} \subset V_{1} \subset \cdots \subset V_{n} = V$ with $\dim V_{i} = i$.

A corollary of Engel's theorem is: Under the hypothesis of the theorem there exists a flag $(V_{i})$ in $V$ stable under $L$, with $x.V_{i} \subset V_{i - 1}$ for all $i$.

The proof of this corollary is as follows: Begin with any nonzero $v \in V$ killed by $L$. The existence of such a $v$ is assured by Engel's Theorem. Set $V_{1} = \mathcal{F}v$ where $\mathcal{F}$ is the ground field. Let $W = V/V_{1}$, and observe that the induced action of $L$ on $W$ is also by nilpotent endomorphisms. By induction on $\dim V$, $W$ has a flag stabilized by $L$ whose inverse image in $V$ is our desired flag.

My question is deals with the last sentence of the above proof. How does the induction exactly work? I also can't seem to see why the inverse image of $W$ in $V$ is the flag we want.

  • 0
    That's just a consequence of the isomorphism theorems: the subspaces of $V/V_1$ are in one-to-one, inclusion preserving correspondence with the subspaces of $V$ that contains $V_1$; so the pullback of the flag in $W$ will give you a chain of subspaces of $V$ that contain $V_1$, with dimensions 1 more than the dimension in $W$.2012-03-25

0 Answers 0