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I'm studying Humphreys' book 'Introduction to Lie Algebras and Representation Theory' First here is theorem 4.1.

Theorem. Let $L$ be a solvable subalgebra of $\mathfrak gl(V)$, $V$ finite dimensional. if $V \neq 0$, then $V$ contains a common eigenvector for all the endomorphisms in $L$.

And here is Corollary A.

Coroallary A. (Lie's Theorem). Let $L$ be a solvable subalgebra of $\mathfrak gl(V)$, dim$V=n \lt \infty$. Then $L$ stabilizes some flag in $V$.

And the book states that we can prove Cor.A. by using the above theorem along with induction on dim$V$. Here is my attempt.

If dim$V$=1, then the only flag is $0 \subset V$ and it is stabilized clearly by $L$. Now assume dim$V=n$. By the main theorem, there exists $v \in V$ such that $x.v=\lambda(x)v$ for all $x\in L$. Then $<v>$ is a subspace which is invariant under $L$. Decompose $V$ as $V=<v> \oplus V'$. Let $L'=L\cap \mathfrak gl(V')$. Clearly, $L'$ is solvable in $\mathfrak gl(V')$. Since dim$V'\lt$ dim$V$, by induction hypothesis, there is a flag of $V'$ stabilized by $L'$.

Here is where I stucked. How can I extend that flag of $V'$ stabilized by $L'$ to a flag of $V$ stabilized by L? I found other questions about this problem, but I still don't get it.

Thanks for any help in advance.

learner
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1 Answers1

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Let $\rho\colon L\rightarrow \mathfrak{gl}(V)$ be a Lie algebra representation of a solvable Lie algebra $L$. We'll use induction over $\dim V$ to show that there is a $L$-invariant flag $$0=V_0\subset V_1 \subset \cdots \subset V_n=V$$ in $V$ such that $\dim V_j=j$. Choosing the basis elements for $V$ as $v_i\in V_i$, the claim follows because of $\rho(x) (V_i)\subset V_i$. For $V=0$ there is nothing to show. So let $\dim V\ge 1$. By the Theorem there is a $v\in V$, $v\neq 0$ with $L.v\subset kv$. It follows that $W=kv$ is a $1$-dimensional $L$-submodule. By applying the induction hypothesis to the quotient module $V/W$ we find there an $L$-invariant flag $0=W_1 \subset \cdots \subset W_n$ with $\dim W_j=j-1$. Let $\pi\colon V\rightarrow V/W$ be the quotient map. Then $V_0=0$ and $V_j=\pi^{-1}(W_j)$ defines a $L$-invariant flag in $V$ with $\dim V_j=j$.

Dietrich Burde
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  • Thanks for your answer. But I still have a question. To apply the induction hypothesis, L should be in \mathfrak gl(V), i.e., L sends V/W into V/W, isn't it? – learner Jan 24 '21 at 06:43
  • Yes, $L$ is assumed to a subalgebra of $\mathfrak{gl}(V)$ in your homework. But in my answer we can chose a faithful $\rho$ so that $\rho(L)\cong L$. And $\rho(L)$ is a linear Lie algebra. – Dietrich Burde Jan 24 '21 at 09:32
  • Oh, I made a typo. I mean, L should be in $\mathfrak gl(V/W)$. not just V. In your answer, can we get L is in gl(V/W)?? – learner Jan 24 '21 at 10:32