Polytopal linear groups

Let $P \subseteq \mathbb{R}^n$ be a lattice polytope; i.e., $P$ is a convex hull of a finite subset of $\mathbb{Z}^n \subseteq \mathbb{R}^n$. Consider the subsemigroup $S_P$ in $\mathbb{Z}^{n+1}$ generated by the set $\{(x; 1) \mid x \in P \cap \mathbb{Z}^n\}$. The polytopal algebra associated with $P$ is the semigroup algebra $k[S_P]$ where $k$ is a field. The algebra $k[S_P]$ is naturally graded by the group $\mathbb{Z}$. Following the work of Winfried Bruns and Joseph Gubeladze we will give a description of the group of graded automorphisms of $k[S_P]$.

References:
[1] Winfried Bruns and Joseph Gubeladze. Polytopal linear groups. Journal of Algebra 218, 715-737 (1999)