Super duality for quantum affine algebras of type $A$

Abstract

We introduce a new approach to the study of finite-dimensional representations of the quantum group of the affine Lie superalgebra $\mathrm{L}{\mathfrak{gl}}_{M|N}=\mathbb{C}[t,t^{-1}]\otimes {\mathfrak{gl}}_{M|N}$ ($M\ne N$). We explain how the representations of the quantum group of $\mathrm{L}{\mathfrak{gl}}_{M|N}$ are directly related to those of the quantum affine algebra of type $A$, using an exact monoidal functor called truncation. This can be viewed as an affine analogue of super duality of type $A$.

59pages, Introduction and Section 6.6 revised, the proof of Theorem 2.9 in the previous version replaced by a new one, the proofs of Lemma 4.15 and Theorem 6.6 revised, comments added on Remarks 5.12 and 6.27 together with minor corrections