The polytope algebra was defined by P. McMullen in "The polytope algebra" Adv. Math. 78 (1989) as follows.

Let us denote by $\Pi'_\mathbb{R}$ the quotient of the free abelian group generated by the symbols $[P]$, where $P\subset \mathbb{R}^n$ is an arbitrary convex compact polytope, by the subgroup generated by elements \begin{eqnarray*} (1)\, [P\cup Q]+[P\cap Q]-[P]-[Q] \mbox{ where } P,Q,P\cup Q \mbox{ are convex compact polytopes};\\ (2)\, [P+x]-[P] \mbox{ where } x\in \mathbb{R}^n. \end{eqnarray*}

Similarly let us define the analogous group $\Pi_{\mathbb{Q}}'$ generated by convex compact polytopes with rational vertices and the same relations (1)-(2).

We have the obvious group homomorphism $\Pi_{\mathbb{Q}}'\to \Pi'_\mathbb{R}$. Is it injective? A reference would be helpful.

The polytope algebra was defined by P. McMullen in "The polytope algebra" Adv. Math. 78 (1989) as follows.

Let us denote by $\Pi'_\mathbb{R}$ the quotient of the free abelian group generated by the symbols $[P]$, where $P\subset \mathbb{R}^n$ is an arbitrary convex compact polytope, by the subgroup generated by elements \begin{eqnarray*} (1)\, [P\cup Q]+[P\cap Q]-[P]-[Q] \mbox{ where } P,Q,P\cup Q \mbox{ are convex compact polytopes};\\ (2)\, [P+x]-[P] \mbox{ where } x\in \mathbb{R}^n. \end{eqnarray*}

Similarly let us define the analogous group $\Pi_{\mathbb{Q}}'$ generated by convex compact polytopes with rational vertices and the same relations (1)-(2) (in the relation (2) one assumes $x\in\mathbb{Q}^n$).

We have the obvious group homomorphism $\Pi_{\mathbb{Q}}'\to \Pi'_\mathbb{R}$. Is it injective? A reference would be helpful.