Quantum groups from homologies of configuration spaces

TL;DR

This paper introduces a method to derive quantum groups and their representations from the twisted homologies of configuration spaces, revealing new combinatorial structures and geometric relations.

Contribution

It reconstructs quantum groups from configuration space homologies, integrating geometric, algebraic, and combinatorial insights in a novel way.

Findings

Quantum groups are reconstructed from configuration space homologies.

New combinatorial structures are introduced in the theory.

Diagrammatic relations correspond to actual homological relations.

Abstract

We reconstruct a quantum group associated with any Lie algebra together with its representation theory from twisted homologies of generalized configuration spaces of disks. Along the way it brings new combinatorics to the theory, but our diagrams represent true submanifolds of configuration spaces and combinatorial relations between them translate actual twisted homological relations.