An Example of the Curvature Tensor for a Quantum Space

TL;DR

This paper introduces the algebra of Q-meromorphic functions on a quantum plane and derives a curvature tensor formula, showing it reduces to flatness in the classical limit as the quantization parameter approaches 1.

Contribution

It provides a new formulation of the curvature tensor for quantum spaces and connects it to classical flatness in the limit of the quantization parameter.

Findings

Derived a formula for the curvature tensor on the algebra of Q-meromorphic functions.

Showed the curvature tensor reduces to flatness as the quantum parameter approaches 1.

Extended the understanding of geometric structures in quantum spaces.

Abstract

The paper is constructed in two parts.In the first part we introduce the concept of the algebra of Q-meromorphic functions on the quantum plane.The A (q)-algebra of Q-analytic functions considered in[6]is seen as a proper subalgebra. In the second part we find a formula for the curvature tensor on this algebra. It is seen that when the quantization parameter tendsto 1,then this formula gives the flatness of the usual R^2 .