|Date||May 03, 2023|
We propose a new approach to the study of representations of quantum affine (super)algebras, based on super duality. Namely, we study a category of interest by finding its bosonic or fermionic counterpart, and then establish supersymmetric analogues and functors interpolate bosons and fermions. A key role is played by R-matrices and their spectral decompositions, which enables a uniform treatment for super and non-super cases.
In this thesis, we consider two module categories of quantum affine (super)algebras of type A. First, the category of polynomial representations is studied, where uniform approach is possible thanks to the powerful Schur-Weyl type duality. We find a direct connection between the category for quantum affine superalgebras and the one for quantum affine algebras by exact monoidal functors, and we lift this to an equivalence between inverse limits of categories.
Second, we introduce a category of infinite dimensional representations called q-oscillator representations. We give a fusion construction of irreducible q-oscillator representations which naturally correspond to finite-dimensional irreducible. Since the former can be seen as a bosonic counterpart of the latter, we introduce an analogous category for quantum affine superalgebras that interpolates the correspondence, which is expected to be explained by a super duality type equivalence.