ScalingStacks

Fix $n\geq 1$. The differential algebra

is the opposite of the strands algebra ${\mathcal{A}}_{LOT}(n)$ with $n$ places of [LiOzTh1, Definition 3.2].

There is a grading on ${\mathcal{A}}_{LOT}(n)$ by a group $G^{\prime}(n)$ [LiOzTh1, §3.3.1]. This gives rise to a grading by $G^{\prime}(n)^{{\operatorname{opp}\nolimits}}$ on ${\mathcal{A}}(n)$.

The group $G^{\prime}(n)^{\operatorname{opp}\nolimits}$ identifies with the index $2$ subgroup $\ker\epsilon\cap\Gamma_{[1,n]^{+}}^{f}$ of $\Gamma_{[1,n]^{+}}^{f}$ via $(r,\alpha)\mapsto(-r,-\alpha)$ (cf Remark 6.2.6 and the identification of $\Gamma_{[1,n]^{+}}$ with the set $\frac{1}{2}{\mathbf{Z}}\times R_{n}$ before Lemma 6.2.7). Via this isomorphism, the $G^{\prime}(n)^{{\operatorname{opp}\nolimits}}$-grading on ${\mathcal{A}}(n)$ comes from our $\Gamma_{[1,n]^{+}}^{f}$-grading on ${\mathcal{H}}_{n}^{f++}$.