A categorification of U_T sl(1,1) and its tensor product representations

Abstract

We define the Hopf superalgebra U_T sl(1,1), which is a variant of the quantum supergroup U_q sl(1,1), and its tensor product 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(A), DGP(B) and DGP(R_n), which are full DG subcategories of the categories of DG A-, B- and R_n-modules generated by certain distinguished projective modules. Their 0th homology categories HP(A), HP(B), and HP(R_n) are triangulated and give algebraic formulations of the contact categories of an annulus, a twice punctured disk, and an n times punctured disk. We categorify the multiplication and comultiplication on U_T sl(1,1) to a bifunctor HP(A) \times HP(A) --> HP(A) and a functor HP(A) --> HP(B), respectively. The U_T sl(1,1)-action on V_1^{\otimes n} is lifted to a bifunctor HP(A) \times HP(R_n) --> HP(R_n).