Hecke-Clifford algebras and spin Hecke algebras II: the rational double affine type

TL;DR

This paper introduces and studies rational spin double affine Hecke algebras and rational double affine Hecke-Clifford algebras, establishing their properties, isomorphisms, and connections to Cherednik algebras in the context of classical Weyl groups.

Contribution

It defines new algebraic structures (sDaHa and DaHCa), proves their fundamental properties, and links them to existing Cherednik algebras, expanding the theoretical framework.

Findings

Established PBW basis and Dunkl operator representations.

Proved an algebra isomorphism between DaHCa and sDaHa.

Linked rational Cherednik algebra to sDaHa via rational covering DAHAs.

Abstract

The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl operator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras.