Dual pairs of quantum moment maps and doubles of Hopf algebras

TL;DR

This paper constructs a natural algebra homomorphism between the Drinfeld double and the Heisenberg double of a finite-dimensional Hopf algebra using quantum moment maps, linking various algebraic structures and their quantizations.

Contribution

It introduces a novel construction of the homomorphism via commuting quantum moment maps, connecting doubles of Hopf algebras and reflection equation algebras.

Findings

Established a homomorphism from D(A) to H(A) using quantum moment maps

Connected the quantization of the Grothendieck-Springer resolution to this framework

Provided insights into the algebraic structures underlying quantum groups

Abstract

For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck-Springer resolution arises in this context.