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

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 $0^{\text{ th}}$ 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 $U_T\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$, $U_T\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right){\otimes }_{\mathbb{Z}}{U}_T\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$ 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 $HP\left(A\right)\times HP\left(A\right)\to HP\left(A\right)$ 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 $HP\left(A\right)\times HP\left(R_n\right)\to HP\left(R_n\right)$.