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A study of nil Hecke algebras via Hopf algebroids

Zbigniew Wojciechowski

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.

A study of nil Hecke algebras via Hopf algebroids

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.

Paper Structure

This paper contains 4 sections, 2 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on bialgebroids and Hopf algebroids
  3. Examples of bialgebroids
  4. Application to nil Hecke algebras

Key Result

Lemma 2.4

The following hold:

Theorems & Definitions (22)

  • Definition 2.3
  • Lemma 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Lemma 2.9
  • proof
  • Example 3.1
  • Example 3.2
  • ...and 12 more