Spectral flow invariants and twisted cyclic theory from the Haar state on SU_q(2)

TL;DR

This paper develops spectral flow invariants for the quantum group SU_q(2) using twisted cyclic theory and K-theoretic methods, revealing complex index computations involving eta contributions.

Contribution

It introduces new index theorems for SU_q(2) employing twisted cyclic cohomology and modular group twisting, extending previous K-theoretic invariants to more typical algebraic settings.

Findings

Index theorems for SU_q(2) using cyclic cohomology

Complex index computations with eta contributions

Application of twisted cyclic theory to quantum groups

Abstract

In [CPR2], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SU_q(2), and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [CPR2], the computations are considerably more complex and interesting, because there are nontrivial `eta' contributions to this index.