ScalingStacks

We define now positive and finite variants of the categories.

We define $\hat{{\mathfrak{S}}}_{n}^{++}$ to be the submonoid of $\hat{{\mathfrak{S}}}_{n}$ of elements $\sigma$ such that $\sigma(r)\geq r$ for all $r\in{\mathbf{Z}}$.

Let $?\in\{+,++\}$. We define ${\mathcal{S}}_{n}^{?}$ to be the $\Gamma_{n}$-filtered subcategory of ${\mathcal{S}}_{n}$ with same objects as ${\mathcal{S}}_{n}$ and with maps those $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ such that $\sigma(r)>0$ if $?=+$ (resp. $\sigma(r)\geq r$ if $?=++$) for all $r\in\tilde{I}\cap{\mathbf{Z}}_{>0}$. We define ${\mathcal{H}}_{n}^{?}$ as the $\Gamma_{n}$-graded pointed subcategory of ${\mathcal{H}}_{n}$ with same objects as ${\mathcal{H}}_{n}$ and non-zero maps those of ${\mathcal{S}}_{n}^{?}$. Note that there is a canonical isomorphism of $\Gamma_{n}$-graded pointed categories ${\operatorname{gr}\nolimits}{\mathcal{S}}_{n}^{?}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{H}}_{n}^{?}$.

Note that the usual symmetric group ${\mathfrak{S}}_{n}$ identifies with the subgroup of $\hat{{\mathfrak{S}}}_{n}$ of elements $\sigma$ such that $\sigma(\{1,\ldots,n\})=\{1,\ldots,n\}$. The subalgebra of $\hat{H}_{n}$ generated by $T_{1},\ldots,T_{n-1}$ is isomorphic to $H_{n}$.

We denote by ${\mathcal{S}}_{n}^{f}$ the $\Gamma_{n}$-filtered subcategory of ${\mathcal{S}}_{n}$ with same objects as ${\mathcal{S}}_{n}$ and with maps those $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ such that $\sigma(r)\in\{1,\ldots,n\}$ for all $r\in\tilde{I}\cap\{1,\ldots,n\}$. We denote by ${\mathcal{H}}_{n}^{f}$ the corresponding $\Gamma_{n}$-graded pointed subcategory of ${\mathcal{H}}_{n}$. There is a canonical isomorphism of $\Gamma_{n}$-graded pointed categories ${\operatorname{gr}\nolimits}{\mathcal{S}}_{n}^{f}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{H}}_{n}^{f}$.

We have also subcategories ${\mathcal{S}}_{n}^{f++}={\mathcal{S}}_{n}^{f}\cap{\mathcal{S}}_{n}^{++}$ of ${\mathcal{S}}_{n}$ and ${\mathcal{H}}_{n}^{f++}={\mathcal{H}}_{n}^{f}\cap{\mathcal{H}}_{n}^{++}$ of ${\mathcal{H}}_{n}$.

We extend all previous constructions to the case $n=0$ by setting $\hat{{\mathfrak{S}}}_{0}=\hat{{\mathfrak{S}}}_{0}^{++}={\mathfrak{S}}_{0}=1$, $H_{0}=\hat{H}_{0}={\mathbf{F}}_{2}$, ${\mathcal{S}}_{0}={\mathcal{S}}_{0}^{++}={\mathcal{S}}_{0}^{f}$ is the category with one object $\emptyset$ and one map and ${\mathcal{H}}_{0}={\mathcal{H}}_{0}^{f}={\mathcal{H}}_{0}^{++}$ is its associated pointed category.

Let $R^{f}_{n}=\bigoplus_{a\in{\mathbf{Z}}/n,\ a\neq-1}{\mathbf{Z}}\alpha_{a}$.

Let $\Gamma^{f}_{n}=\{(r,(l,\alpha))\ |\ \alpha\in R^{f}_{n}\}$, a subgroup of $\Gamma_{n}$. Given $D$ as above, we denote by $\Gamma_{D}^{f}$ the image of $\Gamma^{f}_{n}$ in $\Gamma_{D}$.

Given $\sigma$ a map in ${\mathcal{S}}_{n}^{f}$, we have $\deg(\sigma)\in\Gamma^{f}_{n}$. This shows that the $\Gamma_{n}$-gradings on ${\mathcal{H}}_{n}^{f}$ and ${\mathcal{H}}_{n}^{f++}$ come from $\Gamma_{n}^{f}$-gradings.

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 ${\mathcal{H}}_{n}^{f++}$.