Braided doubles

TL;DR

Braided doubles unify various algebraic structures like quantum groups and Cherednik algebras through a deformation framework involving quasi-Yetter-Drinfeld modules and braidings, revealing new algebraic relationships.

Contribution

The paper introduces braided doubles as a unifying algebraic framework connecting classical and quantum algebras via deformation theory and quasi-Yetter-Drinfeld modules.

Findings

Construction of quantum versions of Weyl algebras using Nichols-Woronowicz algebras.

Identification of rational Cherednik algebras as subalgebras within braided doubles.

Extension of the framework to include classical Yang-Baxter solutions for nonzero parameters.

Abstract

Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. They are a class of algebras with triangular decomposition, arising from a deformation problem, the solutions to which are called quasi-Yetter-Drinfeld modules. A basic family of quasi-YD modules is provided by braidings (matrices satisfying the quantum Yang-Baxter equation); these give rise to quantum versions of the Weyl algebra, where the role of polynomial rings is played by Nichols-Woronowicz algebras. Rational Cherednik algebras for t = 0 emerge as subalgebras in doubles of Nichols-Woronowicz algebras. For nonzero t, the Nichols-Woronowicz algebra is replaced with an algebra associated to the classical Yang-Baxter equation.