A study of nil Hecke algebras via Hopf algebroids

TL;DR

This paper explores how affine nil Hecke algebras can be structured as Hopf algebroids, expanding understanding of their algebraic properties and providing new examples across various mathematical fields.

Contribution

It demonstrates that the affine nil Hecke algebra forms a Hopf algebroid without an antipode using Kostant and Kumar's comultiplication, with new examples included.

Findings

Affine nil Hecke algebra can be structured as a Hopf algebroid

The structure relies on mixed dihedral braid relations

Includes new examples from ring theory, representation theory, and algebraic geometry

Abstract

Hopf algebroids are generalizations of Hopf algebras to less commutative settings. We show how the comultiplication defined by Kostant and Kumar turns the affine nil Hecke algebra associated to a Coxeter system into a Hopf algebroid without an antipode. The proof relies on mixed dihedral braid relations between Demazure operators and simple reflections. For researchers new to Hopf algebroids we include additional examples from ring theory, representation theory, and algebraic geometry.