# Twisted Hochschild homology of quantum flag manifolds: 2-cycles from invariant projections

## TL;DR

This paper investigates the twisted Hochschild homology of quantum flag manifolds, revealing infinite-dimensional 2-cycle spaces for higher rank cases and constructing non-trivial 2-cycles from invariant projections.

## Contribution

It demonstrates that the second twisted Hochschild homology is infinite-dimensional for quantum flag manifolds of rank greater than one and constructs explicit 2-cycles from invariant projections.

## Key findings

- HH_2^θ is infinite-dimensional for rank > 1
- Non-trivial 2-cycles can be constructed from invariant projections
- Discusses quantum Grassmannians as examples

## Abstract

We study the twisted Hochschild homology of quantum full flag manifolds, with the twist being the modular automorphism of the Haar state. We show that non-trivial 2-cycles can be constructed from appropriate invariant projections. The main result is that $HH_2^\theta(\mathbb{C}_q[G / T])$ is infinite-dimensional when $\mathrm{rank}(\mathfrak{g}) > 1$. We also discuss the case of generalized flag manifolds and present the example of quantum Grassmannians.