Braided Weyl algebras and differential calculus on U(u(2))

TL;DR

This paper develops a braided analog of the Weyl algebra on Reflection Equation algebras, introduces partial derivatives, eigenfunctions, and a Laplace operator on the algebra U(u(2)), and explores their properties and potential applications.

Contribution

It constructs a new braided Weyl algebra framework on Reflection Equation algebras and defines differential calculus tools on U(u(2)), extending classical concepts to a quantum algebra setting.

Findings

Defined partial derivatives on U(u(2))

Introduced eigenfunctions and Laplace operator analogs

Expressed the radial part in terms of quantum eigenvalues

Abstract

On any Reflection Equation algebra corresponding to a skew-invertible Hecke symmetry (i.e. a special type solution of the Quantum Yang-Baxter Equation) we define analogs of the partial derivatives. Together with elements of the initial Reflection Equation algebra they generate a "braided analog" of the Weyl algebra. When $q\to 1$, the braided Weyl algebra corresponding to the Quantum Group $U_q(sl(2))$ goes to the Weyl algebra defined on the algebra $\Sym((u(2))$ or that $U(u(2))$ depending on the way of passing to the limit. Thus, we define partial derivatives on the algebra $U(u(2))$, find their "eigenfunctions", and introduce an analog of the Laplace operator on this algebra. Also, we define the "radial part" of this operator, express it in terms of "quantum eigenvalues", and sketch an analog of the de Rham complex on the algebra $U(u(2))$. Eventual applications of our approach are discussed.