  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.
