  This paper aims to show that a certain moduli space $\textsf{T}$, which
arises from the so-called Dwork family of Calabi-Yau $n$-folds, carries a
special complex Lie algebra containing a copy of $\mathfrak{sl}_2(\mathbb{C})$.
In order to achieve this goal, we introduce an algebraic group $\sf G$ acting
from the right on $\textsf{T}$ and describe its Lie algebra ${\rm Lie}({\sf
G})$. We observe that ${\rm Lie}({\sf G})$ is isomorphic to a Lie subalgebra of
the space of the vector fields on $\textsf{T}$. In this way, it turns out that
${\rm Lie}({\sf G})$ and the modular vector field ${\sf R}$ generate another
Lie algebra $\mathfrak{G}$, called AMSY-Lie algebra, satisfying $\dim
(\mathfrak{G})=\dim (\textsf{T})$. We find a copy of
$\mathfrak{sl}_2(\mathbb{C})$ containing ${\sf R}$ as a Lie subalgebra of
$\mathfrak{G}$. The proofs are based on an algebraic method calling
"Gauss-Manin connection in disguise". Some explicit examples for $n=1,2,3,4$
are stated as well.
