Deformations of nilHecke algebras
type: Item
instance of: scholarly article
title: Deformations of nilHecke algebras
zbMATH Open document ID: 1333.55005
author: Benjamin Cooper
published in: Acta Mathematica Vietnamica
publication date: 2015-01-21T00:00:00Z
review text: A nilHecke algebra \(NH_n\) is the graded ring generated by operators \(x_i\) in degree 2 and operators \(\partial_j\) in degree \(-2\), \(1\leq i, j\leq n\) subject to the relations \(\partial_i^2=0\), \(\partial_i \partial_{i+1} \partial_i =\partial_{i+1} \partial_i \partial_{i+1}\), \(x_i\partial_{i}-\partial_{i+1}x_{i+1}=1\), \(\partial_{i}x_i-x_{i+1}\partial_{i}=1\). The algebra \(NH_n\) acts on the polynomial ring \(H^*(BT^n)=\mathbb Z[x_1, \ldots , x_n]\) by the divided difference operators \[ \partial_{i}f=\frac{f-\sigma_i(f)}{x_i-x_{i+1}}, \quad f\in H^*(BT^n) \] where \(\sigma_i\) is the transposition \((j, j+1)\). Now, given a formal group law \(F(x,y)\in Rx,y\), the author defines the generalized nilHecke algebra \(GNH_n=R\otimes NH_n\) by setting \[ \partial_{i}f=\frac{f-\sigma_i(f)}{x_i-_F x_{i+1}}, \quad f\in H^*(BT^n) \] The main result (Theorem 3.11) gives an explicit presentation of the algebra \(GNH_n\). Furthermore, the author constructs a family of deformations of \(GNH's\).
Mathematics Subject Classification ID: 55N22
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 6388892
zbMATH Keywords: complex oriented cohomology
zbMATH Keywords: deformation theory
zbMATH Keywords: formal groups
zbMATH Keywords: Hecke algebra
zbMATH Keywords: nil Hecke algebra
reviewed by: Yuli Rudyak
MaRDI profile type: Publication
full work available at URL: s40306-014-0085-9
OpenAlex ID: W1965726777
cites work: Q4067782
cites work: Steenrod structures on categorified quantum groups
cites work: Characteristic Classes and Homogeneous Spaces, I
cites work: The Schubert Calculus, Braid Relations, and Generalized Cohomology
cites work: Schubert Calculus in Complex Cobordism
cites work: SCHUBERT CELLS AND COHOMOLOGY OF THE SPACESG/P
cites work: Invariants symétriques entiers des groupes de Weyl et torsion
cites work: A diagrammatic approach to categorification of quantum groups I
cites work: Homological Properties of Comodules Over MU ∗ (MU) and BP ∗ (BP)
cites work: Tour of bordered Floer theory
cites work: Q3600688
cites work: Elementary proofs of some results of cobordism theory using Steenrod operations
cites work: Q3901604
cites work: Quelques propriétés globales des variétés différentiables
cites work: Canonical bases and KLR-algebras
DOI: 10.1007/S40306-014-0085-9
Recommended article: Q3960991
Recommended article: Degenerations of nilpotent algebras
Recommended article: Cohomologies et deformations de certames algebres de lie nilpotentes
Recommended article: On Deformations of n-Lie Algebras
Recommended article: Q3926763
Recommended article: Q3709431
Recommended article: Q4237193
Recommended article: On deformations of the Witt n-algebra
Recommended article: Q4318920
description: scientific article; zbMATH DE number 6388892