# Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces

## TL;DR

This paper develops a bicovariant differential calculus framework for noncommutative spaces derived from Lie algebras, providing explicit realizations and extending classical calculus to these quantum-like geometries.

## Contribution

It constructs a Lie superalgebra and explicit realizations for noncommutative coordinates and forms, enabling a deformation of standard calculus on Euclidean space.

## Key findings

- Constructed a Lie superalgebra for noncommutative spaces
- Provided realizations as formal power series in Weyl superalgebra
- Established a bicovariant differential calculus as a deformation

## Abstract

This paper investigates bicovariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra $g_0$ we construct a Lie superalgebra $g=g_0\oplus g_1$ containing noncommutative coordinates and one--forms. We show that $g$ can be extended by a set of generators $T_{AB}$ whose action on the enveloping algebra $U(g)$ gives the commutation relations between monomials in $U(g_0)$ and one--forms. Realizations of noncommutative coordinates, one--forms and the generators $T_{AB}$ as formal power series in a semicompleted Weyl superalgebra are found. In the special case $dim(g_0)= dim(g_1)$ we also find a realization of the exterior derivative on $U(g_0)$. The realizations of these geometric objects yield a bicovariant differential calculus on $U(g_0)$ as a deformation of the standard calculus on the Euclidean space.