#### Volume 18, issue 3 (2014)

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A categorification of $\boldsymbol{U}_T(\mathfrak{sl}(1|1))$ and its tensor product representations

### Yin Tian

Geometry & Topology 18 (2014) 1635–1717
##### Abstract

We define the Hopf superalgebra ${U}_{T}\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$, which is a variant of the quantum supergroup ${U}_{q}\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$, and its representations ${V}_{1}^{\otimes n}$ for $n>0$. We construct families of DG algebras $A$, $B$ and ${R}_{n}$, and consider the DG categories $DGP\left(A\right)$, $DGP\left(B\right)$ and $DGP\left({R}_{n}\right)$, which are full DG subcategories of the categories of DG $A$–, $B$– and ${R}_{n}$–modules generated by certain distinguished projective modules. Their homology categories $HP\left(A\right)$, $HP\left(B\right)$ and $HP\left({R}_{n}\right)$ are triangulated and give algebraic formulations of the contact categories of an annulus, a twice punctured disk and an $n$ times punctured disk. Their Grothendieck groups are isomorphic to ${U}_{T}\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$, ${U}_{T}\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right){\otimes }_{ℤ}{U}_{T}\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$ and ${V}_{1}^{\otimes n}$, respectively. We categorify the multiplication and comultiplication on ${U}_{T}\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$ to a bifunctor $HP\left(A\right)×HP\left(A\right)\to HP\left(A\right)$ and a functor $HP\left(A\right)\to HP\left(B\right)$, respectively. The ${U}_{T}\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$–action on ${V}_{1}^{\otimes n}$ is lifted to a bifunctor $HP\left(A\right)×HP\left({R}_{n}\right)\to HP\left({R}_{n}\right)$.

##### Keywords
Hopf superalgebra, categorification, tight contact structure, Heegaard Floer homology
##### Mathematical Subject Classification 2010
Primary: 18D10
Secondary: 16D20, 57M50