The Two-Parameter Higher Order Differential Calculus and Curvature on a Quantum Plane

TL;DR

This paper develops a two-parameter differential calculus on a quantum plane, deriving curvature and formulating a non-commutative gauge theory, expanding the mathematical framework of quantum geometry.

Contribution

It introduces a novel two-parameter differential calculus on a quantum plane with nilpotent endomorphisms and constructs associated curvature and gauge theories.

Findings

Constructed associative differential algebra for d^2=0 and d^3=0 cases.

Derived curvature in the non-commutative setting.

Formulated a non-commutative gauge field theory.

Abstract

We construct an associative differential algebra on a two-parameter quantum plane associated with a nilpotent endomorphism $d$ in the two cases $d^{2}=0$ and $d^3=0$ $(d^2\neq 0).$ The correspondent curvature is derived and the related non commutative gauge field theory is introduced.