Quantum and Braided Integrals

TL;DR

This paper introduces integration methods for non-commutative spaces using Hopf algebra frameworks, including new trace formulas and braided algebra techniques, supported by detailed examples and pedagogical explanations.

Contribution

It provides a comprehensive pedagogical overview of integration in non-commutative geometry, introducing new trace formulas and extending to braided Hopf algebras with diagrammatic methods.

Findings

Derived new trace formulas for Hopf algebra integrals

Extended integration techniques to braided Hopf algebras

Provided explicit worked examples illustrating the methods

Abstract

We give a pedagogical introduction to integration techniques appropriate for non-commutative spaces while presenting some new results as well. A rather detailed discussion outlines the motivation for adopting the Hopf algebra language. We then present some trace formulas for the integral on Hopf algebras and show how to treat the $\int 1=0$ case. We extend the discussion to braided Hopf algebras relying on diagrammatic techniques. The use of the general formulas is illustrated by explicitly worked out examples.