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