Invariants of tangles with flat connections in their complements

TL;DR

This paper introduces invariants for tangles with flat G-connections in their complements using G-categories, and demonstrates that quantized universal enveloping algebras at roots of unity serve as examples of such categories.

Contribution

It defines a new class of invariants for tangles with flat connections and links them to quantized universal enveloping algebras at roots of unity.

Findings

Invariants are constructed using G-categories.

Quantized universal enveloping algebras at roots of unity are examples of G-categories.

The framework generalizes previous tangle invariants to include flat G-connections.

Abstract

Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide examples of G-categories.