  We classify finite dimensional $H_{m^2}(\zeta)$-simple
$H_{m^2}(\zeta)$-module Lie algebras $L$ over an algebraically closed field of
characteristic $0$ where $H_{m^2}(\zeta)$ is the $m$th Taft algebra. As an
application, we show that despite the fact that $L$ can be non-semisimple in
ordinary sense, $\lim_{n\to\infty}\sqrt[n]{c_n^{H_{m^2}(\zeta)}(L)} = \dim L$
where $c_n^{H_{m^2}(\zeta)}(L)$ is the codimension sequence of polynomial
$H_{m^2}(\zeta)$-identities of $L$. In particular, the analog of Amitsur's
conjecture holds for $c_n^{H_{m^2}(\zeta)}(L)$.
