Questions about Heisenberg lie algebra and it's representation.

Question

I am learning the representation of lie algebra and I have met the following question with no clues...Can anyone help me to answer or give me a hint?

The question is as follows:

The the matrix realization of Heisenberg Lie algebra is $$H= \bigg\{ \begin{bmatrix} 0&x&y \\ 0&0&z \\ 0&0&0 \end{bmatrix}: \ x,y,z \in F\bigg\}$$

Then consider $V=F^3$ as a representation of $H$, where the action of any element x $\in H $on any vector $v \in V $ is by matrix multiplication.

Then prove than $V$ has a unique maximal proper $H$-submodule $W$. And what is $V/W$ as a representation of $H$?

Thank you in advance!

Answer 1

For each $v \in V$, we consider the $H$-submodule of $V$ generated by $v$ - let's call this submodule $M_v$. In general, for $H$ an arbitrary Lie algebra and $V$ an arbitrary $H$-module, $H$ doesn't contain an identity and isn't necessarily closed under composition of the module action, so this object looks like $$M_v = Fv + Hv + H(H v) + H(H(H(v))) + \cdots.$$ In this case $V$ is the defining representation of $H$ and $H$ does happen to be closed under composition of its action on $V$, so we just have $M_v = Fv+Hv$.

Note that every submodule of $V$ can be written as $M_{v_1} + \cdots + M_{v_k}$ for some $v_1, \ldots, v_k \in V$, so to understand the structure of all submodules of $V$, it is enough to understand the structure of those generated by one element.

If we write $$v = \begin{bmatrix}a \\ b \\ c\end{bmatrix},$$ then we can compute $Fv+Hv$ for each possible $v$:

• $c \neq 0$. Then $M_v = V$.
• $c = 0$, but $b \neq 0$. Then $M_v = \left\{\begin{bmatrix}x \\ y \\ 0\end{bmatrix} : x, y \in F\right\}$.
• $c = 0$ and $b=0$, but $a \neq 0$. Then $M_v = \left\{\begin{bmatrix}x \\ 0 \\ 0\end{bmatrix} : x \in F\right\}$.
• $a=b=c=0$. Then $M_v = \{0\}$.

Observe that the submodules generated by one element are nested inside each other, so they are all of the submodules of $V$. And the largest of them that is not equal to $V$ is the unique maximal submodule of $V$. I'll leave it to you to compute the action on the quotient space $V/W$.