Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 3

Special Issue: In Honor of Prof. Kyoji Saito’s 75th Birthday

Guest Editors: Stanislaw Janeczko, Si Li, Jie Xiao, Stephen S.T. Yau, and Huaiqing Zuo

Shuffle algebras and perverse sheaves

Pages: 573 – 657

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n3.a9


Mikhail Kapranov (Kavli IPMU, Kashiwa, Chiba, Japan)

Vadim Schechtman (Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France)


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 equivalence, 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.

The research of M.K. was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and by the IAS School of Mathematics.

Received 2 April 2019

Accepted 4 January 2020

Published 11 November 2020