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The $n$-th Heisenberg algebra $H_n$ over $\mathbb{C}$ is the Lie algebra with basis $\delta_1,\cdots,\delta_n,b_1,\cdots,b_n,c$ and Lie brackets $[\delta_i,\delta_j]=0$, $[b_i,b_j]=0$, $[\delta_i,b_i]=c$, $[\delta_i,b_j]=0$ for $i\neq j$, $[c,x]=0$ for all $x\in H_n$. Prove that $H_n$ has a unique representation $V$ up to isomorphism satisfying three properties:(1)there is $v\in V$, $v\neq 0$ s.t. $\delta_iv=0$;(2)$c$ acts as $1$;(3)$V$ is irreducible.

To model the Lie bracket, I want to consider the matrix $\left(\begin{array}{ccc}0&\textbf{a}^T&c\\0&\textbf{0}&\textbf{b}\\0&\textbf{0}&0\end{array}\right)$, where $\textbf{a}^T=(\delta_1,\cdots,\delta_n)$ and $\textbf{b}^T=(b_1,\cdots,b_n)$. Then an element in $H_n$ can be expressed by a matrix of this form. For $A,B\in H_n$(We automatically see $A,B$ as matrices), we define $[A,B]=AB-BA$. Then the new bracket is nothing but the Lie bracket in the problem. A natural thought is to look for a $V$ a subspace of $\mathbb{C}^{n+2}$. But I fail. I also don't know how to prove the uniqueness.

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    Imagine you have a representation with a nonzero vector $c$ such that $\delta_iv=0$ for all $i$ and $cv=v$. What other vectors in $V$ can you produce from $v$ and the action of the Lie algebra? – 2017-02-02
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    Can Mariano Suárez-Álvarez explain a bit more? – 2017-02-12
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    Li Li, can you think of a column vector that is a common eigenvector to all the matrices of that shape? – 2017-02-12
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    I have obtained the answer along the path given by Mariano Suarez-Alvarez. But I am too lazy to type it into computer.-_-! – 2017-02-17

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