Higher Tensor Product for sl2 and Webster algebras

TL;DR

The work builds a concrete, $t$-structure–aware model for tensoring the regular $2$-representation of $\mathfrak{sl}_2^+$ with the vector $2$-representation, using an infinite generating family to obtain a simpler, yet equivalent, description to Webster's tensor product category. It develops a detailed left and right action of the monoidal category ${\mathcal U}$, analyzes the underlying abelian and derived categories, and constructs explicit complexes $Y_n$ and $Y_{n,m}$ that generate the heart and realize a braided tensor-product structure. A key contribution is the explicit comparison with Webster algebras via graded bimodule functors, establishing a graded, faithful equivalence that aligns the two frameworks. The results provide a tractable, graded categorification of the $U_q(\mathfrak{sl}_2)$–style tensor product in the setting of $2$-representations, contributing to Crane–Frenkel’s program for braided monoidal categories of higher representations.

Abstract

We construct a model for the tensor product of the regular 2-representation of the enveloping algebra of $\mathfrak{sl}_2^+$ with the vector 2-representation, based on the $\infty$-categorical definition of the second author. Our model contains McMillan's minimal one. Our use of an infinite family of generators provides a simpler model that we prove is equivalent to Webster's tensor product category.

Theorems & Definitions (34)

• Remark 4.1
• Proposition 4.2
• proof
• Lemma 5.1
• proof
• Lemma 5.2
• proof
• Lemma 5.3
• proof
• Lemma 5.4