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This is a followup to my previous question here.

Let $\text{U}_+$ be an associative $\mathbb{C}$-algebra with two generators $E$, $H$, and one defining relation $HE - EH = 2E$. Let $M$ be a finite-dimensional $\text{U}_+$-module. Does there exist a nonzero eigenvector $v \in M$, of $H$, such that $E(v) = 0$?

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1 Answers1

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Here is the important part. Let $v $ be an eigenvector for $h $. Then

$$ hev = (2e+eh)v = (\lambda +2)ev, $$

so $ev $ is also an eigenvector for $h $ with eigenvalue $(\lambda+2) $. But $M $ is finite dimensional. Can you complete the argument?

Max
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