  Let $\frak g$ be a simple finite-dimensional Lie algebra over an
algebraically closed field $\mathbb F$ of characteristic 0. We denote by
$\operatorname{U}(\frak g)$ the universal enveloping algebra of $\frak g$. To
any nilpotent element $e\in \frak g$ one can attach an associative (and
noncommutative as a general rule) algebra $\operatorname{U}({\frak g},~e)$
which is in a proper sense a "tensor factor" of $\operatorname{U}(\frak g)$. In
this article we consider the case in which $\frak g$ is simple and $e$ belongs
of the minimal nonzero nilpotent orbit of $\frak g$. Under these assumptions
$\operatorname{U}({\frak g}, e)$ was described explicitly in terms of
generators and relations. One can expect that the representation theory of
$\operatorname{U}({\frak g}, e)$ would be very similar to the representation
theory of $\operatorname{U}(\frak g)$. For example one can guess that the
category of finite-dimensional $\operatorname{U}({\frak g}, e)$-modules is
semisimple.
  The goal of this article is to show that this is the case if $\frak g$ is not
simply-laced. We also show that, if $\frak g$ is simply-laced and is not of
type $A_n$, then the regular block of finite-dimensional
$\operatorname{U}({\frak g}, e)$-modules is equivalent to the category of
finite-dimensional modules of a zigzag algebra.
