# The Plancherel formula for complex semisimple quantum groups

## TL;DR

This paper derives the Plancherel formula for complex semisimple quantum groups, providing a detailed description of their reduced group C*-algebras using advanced algebraic tools.

## Contribution

It computes the Plancherel formula for Drinfeld doubles of q-deformed compact semisimple Lie groups, a novel result in quantum group analysis.

## Key findings

- Explicit Plancherel formula for complex semisimple quantum groups
- Concrete description of associated reduced group C*-algebras
- Application of Bernstein-Gelfand-Gelfand complex and Hopf trace formula

## Abstract

We calculate the Plancherel formula for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. As a consequence we obtain a concrete description of their associated reduced group $ C^* $-algebras. The main ingredients in our proof are the Bernstein-Gelfand-Gelfand complex and the Hopf trace formula.