Quantum Orbifolds

TL;DR

This paper develops a generalized framework for constructing quantum orbifolds from compact quantum groups, analyzing their algebraic structures and spectral triples, and applying these to a Hopf-equivariant Fredholm index problem.

Contribution

It introduces a new mechanism to build quantum orbifolds from any compact simple quantum group, extending previous studies beyond quantum SU(2).

Findings

Constructed quantum orbifolds from general compact quantum groups.

Established spectral triples over invariant subalgebras and crossed product algebras.

Applied to a Hopf-equivariant Fredholm index problem.

Abstract

This is a study of orbifold-quotients of quantum groups (quantum orbifolds $\Theta \rightrightarrows G_q$). These structures have been studied extensively in the case of the quantum $SU_2$ group. I will introduce a generalized mechanism which allows one to construct quantum orbifolds from any compact simple and simply connected quantum group. Associated with a quantum orbifold there is an invariant subalgebra as well as a crossed product algebra. For each spin quantum orbifold, there is a unitary equivalence class of Dirac spectral triples over the invariant subalgebra, and for each effective spin quantum orbifold associated with a finite group action, there is a unitary equivalence class of Dirac spectral triples over the crossed product algebra. As an application I will study a Hopf-equivariant Fredholm index problem.