# Invariance and the twisted Chern character : a case study

## TL;DR

This paper investigates the invariance properties of the Chern character in quantum groups, proving non-invariance under certain coactions and confirming the nontriviality of a twisted Chern character from an equivariant spectral triple on SU_q(2).

## Contribution

It provides a detailed proof that the Chern characters of canonical generators on the K-homology of SU_q(2) are not invariant under the natural coaction and confirms the nontriviality of a specific twisted Chern character.

## Key findings

- Chern characters are not invariant under SU_q(2) coaction.
- The twisted Chern character from an odd equivariant spectral triple is nontrivial.
- The conjecture about the nontriviality of the twisted Chern character is affirmed.

## Abstract

We give details of the proof of the remark made in \cite{G2} that the Chern characters of the canonical generators on the K homology of the quantum group $SU_q(2)$ are not invariant under the natural $SU_q(2)$ coaction. Furthermore, the conjecture made in \cite{G2} about the nontriviality of the twisted Chern character coming from an odd equivariant spectral triple on $SU_q(2)$ is settled in the affirmative.