Author(s): Dimitri Gurevich, Pavel Pyatov
On any Reflection Equation algebra corresponding to a skew-invert ible 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 → 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 th e ”radial part” of this operator, express it in terms of ”quantum eigenvalues”, and sket ch an analog of the de Rham complex on the algebra U ( u (2)). Eventual applications of our approach are discussed.