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

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→1q→1, the braided Weyl algebra corresponding to the Quantum Group Uq(sl(2))Uq(sl(2)) goes to the Weyl algebra defined on the algebra Sym(u(2)) or U(u(2))U(u(2)) depending on the way of passing to the limit. Thus, we define partial derivatives on the algebra U(u(2))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))U(u(2)). Eventual applications of our approach are discussed.