# Hecke-Hopf algebras

## TL;DR

This paper introduces new Hopf algebras called Hecke-Hopf algebras that contain Hecke algebras as coideal subalgebras, offering solutions to the quantum Yang-Baxter equation and new functors for module categories.

## Contribution

It constructs Hecke-Hopf algebras for Coxeter groups, generalizing known structures and connecting to Nichols algebras and braided derivatives.

## Key findings

- Provides new solutions to quantum Yang-Baxter equation
- Constructs a family of endo-functors for module categories
- Relates to Fomin-Kirillov algebras and Nichols algebras

## Abstract

Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{\bf q}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras ${\bf H}(W)$ have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of $H_{\bf q}(W)$-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras, for an arbitrary Coxeter group $W$ the "Demazure" part of ${\bf H}(W)$ is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra.