ScalingStacks

Remark 5.4.1. As in Remark 4.4.3, we obtain a lax bi-$2$-representation on $B$ by setting $E_{i,j}=E_{2}^{j}F_{1}^{i}$. We have an injective morphism of differential algebras $\Delta_{E}(B)\to\Delta_{\lambda}(B)$.

Assume the morphisms (5.2.1) are isomorphisms for all $i$ (this holds for example if $\lambda$ is an isomorphism). Then we have a canonical isomorphism $\Delta_{E}(B)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\Delta_{\lambda}(B)$. The algebra $\Delta_{\lambda}(B)$ is generated by $B$ and $E_{2}F_{1}$.