Cremmer-Gervais r-matrices and the Cherednik Algebras of type GL2

TL;DR

This paper interprets Cremmer-Gervais r-matrices within Cherednik and double affine Hecke algebras, computing their nilpotency index and linking quantization to algebraic structures.

Contribution

It provides a novel interpretation of Cremmer-Gervais r-matrices using Cherednik and Hecke algebras, and calculates their nilpotency index.

Findings

Computed the nilpotency index of Jordanian r-matrices.

Linked Cremmer-Gervais quantization to double affine Hecke algebra.

Provided algebraic interpretations of r-matrices in type GL2.

Abstract

We give an intepretation of the Cremmer-Gervais r-matrices for sl(n) in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL2 on certain subspaces of their polynomial representations. This is used to compute the nilpotency index of the Jordanian r-matrices, thus answering a question of Gerstenhaber and Giaquinto. We also give an interpretation of the Cremmer-Gervais quantization in terms of the corresponding double affine Hecke algebra.