# Representation of Heisenberg Algebra

I tried one problem from "Introduction to Lie Algebras” by Karin Erdmann and Mark Wildon and there is one question that I am not sure how to do.

"Let $L$ be a Heisenberg lie algebra with basis $f,g,z$ such that $[f,g]=z$, and z lies in the center of $L$. Show that there is no finite dimensional faithful irreducible representation of L."

My attempt so far is as follows:

Assume that $V$ is a finite dimensional faithful irreducible representation of $L$. So there is an injective lie algebra homomorphism $\phi:L\rightarrow gl(V)$. It is easy to check that the set $z\cdot V=\{\phi(z)(v)|v\in V\}$ is an $L$ submodule of $V$. Since $V$ is irreducible representation of $L$, it follows that either $z\cdot V=\{0\}$ or $z\cdot V=V$. However, $z\cdot V\neq\{0\}$ because $\phi$ is an injective map. Therefore, $z\cdot V=V$.

I am not sure how to continue from here. Are there any hints how to proceed from here?

• I think the book you refer to is the one by Karin Erdmann and Mark Wildon? I’m sure you didn’t intend any disrespect, but it seemed odd to omit one author and refer to the other just by her first name, so I’ve edited your question. Commented Mar 3, 2018 at 19:47

Since the Heisenberg Lie algebra is nilpotent, it is also solvable. By Lie's theorem, every finite-dimensional irreducible representation of a solvable Lie algebra is $1$-dimensional. On the other hand, the minimal dimension of a faithful representation of the Heisenberg Lie algebra is $3$. Hence there is no finite-dimensional faithful irreducible representation of the Heisenberg Lie algebra.
• You can first apply Lie's theorem over an algebraically closure, and then apply the "Lefschetz's principle" sometimes. On the other hand, Erdmann and Wilson anyway assume $K=\mathbb{C}$, right? Commented Mar 4, 2018 at 19:54