ScalingStacks

The category $\Rep(U_{q}(\mathfrak{gl}_{N}))$ of finite-dimensional $U_{q}(\mathfrak{gl}_{N})$-modules is a ribbon category and thus provides Reshetikhin–Turaev invariants of framed oriented tangles with components labeled by objects of $\Rep(U_{q}(\mathfrak{gl}_{N}))$. A framed oriented link $L$ labeled by the $U_{q}(\mathfrak{gl}_{N})$-module $V={\mathbb{C}}^{N}(q)$ yields an endomorphism of ${\mathbb{C}}(q)$, the tensor unit in $\Rep(U_{q}(\mathfrak{gl}_{N}))$, which is just multiplication by the $\mathfrak{gl}_{N}$ link polynomial of $L$.

While we will focus on invariants of links labeled by $V$, it is convenient to also consider the fundamental modules $\bigwedge^{k}(V)$ and their duals. Together, these generate the full monoidal subcategory $\mathrm{Fund}(U_{q}(\mathfrak{gl}_{N}))$, which admits a graphical presentation and which recovers $\Rep(U_{q}(\mathfrak{gl}_{N}))$ upon idempotent completion.

The ${\mathbb{C}}(q)$-linear pivotal category $\boldsymbol{\mathrm{Web}}_{N}$ has objects given by finite sets of points in an interval ${[0,1]}$, each labeled by an element of $\{n,n^{*}|n\in{\mathbb{Z}}_{>0}\}$. The morphisms are ${\mathbb{C}}(q)$-linear combinations of webs: oriented trivalent graphs, properly embedded in ${[0,1]}^{2}$, with edges labeled by a non-negative integer flow, considered up to isotopy relative to the boundary and certain local relations, including those shown in (2.1). The source and target of a web are determined by its intersections with ${[0,1]}\times\{0\}$ and ${[0,1]}\times\{1\}$, with downward oriented boundary points of label $n$ being recorded as $n^{*}$. Composition is given by the bilinear extension of stacking webs and the tensor product is given on objects by concatenating labeled intervals and on morphisms by the bilinear extension of placing webs side by side.

The morphisms in $\boldsymbol{\mathrm{Web}}_{N}$ are generated under composition, tensor product, and duality by identity morphisms and trivalent merge and split vertices:

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The merge and split vertices encode the natural $U_{q}(\mathfrak{gl}_{N})$-intertwiners

$$\textstyle\bigwedge^{a}(V)\otimes\bigwedge^{b}(V)\to\bigwedge^{a+b}(V)\quad\text{and}\quad\bigwedge^{a+b}(V)\to\bigwedge^{a}(V)\otimes\bigwedge^{b}(V),$$

respectively. The local relations in $\boldsymbol{\mathrm{Web}}_{N}$ include

(2.1)

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together with the reflections of these relations in a vertical line. Edges labeled zero are to be erased and edges labeled by negative integers or integers greater than $N$ force the morphism to be the zero morphism. The local relations ensure that $\boldsymbol{\mathrm{Web}}_{N}\cong\mathrm{Fund}(U_{q}(\mathfrak{gl}_{N}))$ as ${\mathbb{C}}(q)$-linear pivotal categories, see [CKM14, QS19, TVW17]. In the following, we also consider an integral version of $\boldsymbol{\mathrm{Web}}_{N}$, which is defined over ${\mathbb{Z}}[q^{\pm 1}]$ and also satisfies the relations (2.1).