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$\space$ Let $F$ be a field and let $L = b(n,F)$ be the Lie algebra of $n×n$ upper triangular matrices and $V = {F^n}$ . Let $e_{1} , ... , e_{n}$ be the standard basis of $F^n$. For $1 \le r \le n$ , $W_{r} = Span \{ e_{1} , ... , e_{n}\}$. Prove that $W_r$ is a submodule of V.

$\space$ Definition (submodule): $\space$ Suppose that $V$ is a Lie module for the Lie algebra $L$. A submodule of $V$ is a subspace $W$ of $V$ which is invariant under the action of L. That is, for each $x \in L$ and for each $w \in W$, we have $x.w \in W$.

I Consider the matrix $n×n$ upper triangular and I applied this matrix to the element $e_2$ of $W_r$, but the result does not belong to $W_r$. How to make? Any suggestion?

Danilo Santana
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1 Answers1

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If $A$ is an upper triangular matrix in the base $(e_1,...,e_n)$, for every $r$, $A(e_r)=a_{rr}e_r+a_{r-1r}e_{r-1}+..a_{r1}e_1$, let $W_r=Span(e_1,...,e_r)$, $A(e_r)$ is thus an element of $W_r$ and for $i\leq r$, $A(e_i)\in W_i\subset W_r$, this implies that $A(W_r)\subset W_r$ and $W_r$ is a submodule of $V$ for the action defined by $L$.

Tsemo Aristide
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