Derivatives in noncommutative calculus and deformation property of quantum algebras

TL;DR

This paper develops noncommutative derivatives and de Rham complexes for certain quantum algebras, and investigates their deformation properties, revealing that the q-Witt algebra lacks a PBW theorem analog.

Contribution

It introduces derivatives and de Rham complexes on noncommutative algebras and analyzes the deformation properties, challenging existing assumptions about the q-Witt algebra.

Findings

Constructed noncommutative derivatives and de Rham complexes.

Showed the q-Witt algebra does not have a PBW theorem analog.

Discussed various Jacobi conditions for quadratic-linear 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", namely, the so-called modified Reflection Equation algebras. By using these 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 theorem. In this connection, we discuss different forms of the Jacobi condition related to quadratic-linear algebras.