# Shuffle algebras and perverse sheaves

## TL;DR

This paper establishes a deep connection between shuffle algebras and perverse sheaves on symmetric products of the complex line, revealing a new geometric perspective on algebraic structures through perverse sheaves and configuration spaces.

## Contribution

It constructs an equivalence between certain braided Hopf algebras and factorizable collections of perverse sheaves on symmetric products, linking algebraic and geometric frameworks.

## Key findings

- Nichols algebra corresponds to intersection cohomology extensions of local systems
- Equivalence between braided Hopf algebras and perverse sheaves collections
- Uses real skeleta of complex configuration spaces for the construction

## Abstract

We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso, to perverse sheaves on symmetric products of the complex line (i.e., on the spaces of monic polynomials stratified by multiplicities of roots). More precisely, we construct an equivalence between: (i) Braided Hopf algebras of a certain type. (ii) Factorizable collections of perverse sheaves on all the symmetric products. Under this eqiuvalence, the Nichols algebra associated to an object V corresponds to the collection of the intersection cohomology extensions of the local systems on the open configuration spaces associated to the tensor powers of V. Our approach is based on using real skeleta of complex configuration spaces.