Derivatives in noncommutative calculus and deformation property of quantum algebras

Abstract

The aim of the paper is twofold. First, we introduce analogs of (partial) derivatives on certain noncommutative algebras, including some enveloping algebras and their “braided counterparts” – the so-called modified Reflection Equation algebras. With the use of the mentioned derivatives we construct an analog of the de Rham complex on these algebras. Second, we discuss deformation property of some quantum algebras and show that contrary to a commonly held view, in the so-called q-Witt algebra there is no analog of the PBW property. In this connection, we discuss different forms of the Jacobi condition related to quadratic-linear algebras.