Hecke-Hopf algebras

Abstract

Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{\mathbf{q}}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras $\mathbf{H}(W)$ 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 $H_{\mathbf{q}}(W)$-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras, for an arbitrary Coxeter group $W$ the "Demazure" part of $\mathbf{H}(W)$ is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra.

AMSLaTex 67 pages, to appear in Advances in Mathematics