Quantum group connections

TL;DR

This paper generalizes the construction of the Ahtekar-Isham C*-algebra in Loop Quantum Gravity by replacing the classical group with a compact quantum group, extending the algebraic framework to non-commutative settings.

Contribution

It introduces a new method to construct quantum group connections, broadening the algebraic tools available in Loop Quantum Gravity.

Findings

Generalized the algebraic construction to compact quantum groups

Provided a framework for non-commutative connections

Extended inductive techniques to quantum group settings

Abstract

The Ahtekar-Isham C*-algebra known from Loop Quantum Gravity is the algebra of continuous functions on the space of (generalized) connections with a compact structure Lie group. The algebra can be constructed by some inductive techniques from the C*-algebra of continuous functions on the group and a family of graphs embedded in the manifold underlying the connections. We generalize the latter construction replacing the commutative C*-algebra of continuous functions on the group by a non-commutative C*-algebra defining a compact quantum group.