# DG-Enhanced Hecke and KLR Algebras

## TL;DR

This paper constructs DG-enhanced versions of affine Hecke and KLR algebras, proving their isomorphisms after completion and showing their homologies are concentrated in degree zero, linking them to cyclotomic Hecke algebras.

## Contribution

It introduces DG-enhanced versions of affine Hecke and KLR algebras and establishes their isomorphisms, extending prior algebraic frameworks.

## Key findings

- DG-enhanced affine Hecke algebras are isomorphic to DG-enhanced KLR algebras after completion.
- Homologies of these DG-algebras are concentrated in degree zero.
- Homologies correspond to cyclotomic Hecke algebras.

## Abstract

We construct DG-enhanced versions of the degenerate affine Hecke algebra and of the affine Hecke algebra. We extend Brundan-Kleshchev and Rouquier's isomorphism and prove that after completion DG-enhanced versions of affine Hecke algebras (degenerate or nondegenerate) are isomorphic to completed DG-enhanced versions of KLR algebras for suitably defined quivers. As a byproduct, we deduce that these DG-algebras have homologies concentrated in degree zero. These homologies are isomorphic respectively to the degenerate cyclotomic Hecke algebra and the cyclotomic Hecke algebra.