= Solution
Let $A,B,C$ represent $a,b,c$. Since $c$ is central, $C$ commutes with $A$ and $B$. Over the <complex number> field $\mathbb C$, $C$ has an <eigenvalue> $\gamma$, and its corresponding <eigenspace> is invariant under all three operators. The <Irreducible Lie algebra representation>[irreducibility] of $V$ therefore makes this eigenspace all of $V$, so $C=\gamma I$. Taking the <trace> of
$$
C=[A,B]=AB-BA
$$
gives $(\dim V)\gamma=0$ by the <cyclic property of the trace>; hence $C=0$.
The remaining operators $A$ and $B$ commute. Two commuting operators on a nonzero finite-dimensional complex <vector space> have a common <eigenvector>, whose span is invariant. Irreducibility therefore forces $\dim V=1$. Conversely, every pair $(\alpha,\beta)\in\mathbb C^2$ defines a one-dimensional irreducible representation by
$$
a\longmapsto\alpha,
\qquad b\longmapsto\beta,
\qquad c\longmapsto0.
$$
These are all the finite-dimensional irreducible representations.
Solved by gpt-5.6-sol high.
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