A categorification of representations of $U_q(\mathfrak{gl}_{1|1})$

TL;DR

This work constructs a geometric categorification of the action of $U_q(\mathfrak{gl}_{1|1})$ on $(\mathbb{C}^{1|1})^{\otimes N}$ via Fourier–Mukai functors between derived categories $\mathcal{D}(\lambda)$ of Calabi–Yau total spaces $Y(\lambda)$ over Grassmannians. It proves that the categorified generators $\mathcal{E}(\lambda)$ and $\mathcal{F}(\lambda)$ satisfy the corresponding nilpotent and cone relations, and that decategorification in localized $\mathbb{C}^*$-equivariant $K$-theory recovers the expected $2^N$-dimensional representation $(\mathbb{C}^{1|1})^{\otimes N}$. The approach hinges on precise FM kernel constructions using determinants of tautological bundles, generalized Koszul complexes, and push-forwards, yielding a coherent algebro-geometric realization of the quantum group action. The authors sketch a path to generalize to $U_q(\mathfrak{gl}_{m|n})$ via flag varieties, Slodowy slices, and related geometric structures, linking to prior work on categorifications and suggesting broader applicability to symmetric powers and other representations.

Abstract

We categorify the action of $U_q(\mathfrak{gl}_{1|1})$ on the tensor product of its vector representations $(\mathbb{C}^{1|1})^{\otimes N}$. The generators $E$ and $F$ are represented by Fourier-Mukai functors between the derived categories of coherent sheaves on the total spaces of "semi-parabolic" vector bundles over the Grassmannians $Gr(k,N)$.

Theorems & Definitions (6)

• Theorem 1
• Corollary 1
• Remark 1
• Proposition 1
• proof
• Corollary 2