# Peter-Weyl bases, preferred deformations, and Schur-Weyl duality

## TL;DR

This paper explores a preferred deformation of the function algebra on a reductive Lie group, utilizing Peter-Weyl bases, and examines its relation to Schur-Weyl duality, with implications for quantum group structures.

## Contribution

It introduces a specific preferred deformation of the function algebra on reductive Lie groups using Peter-Weyl bases and connects it to Schur-Weyl duality.

## Key findings

- The deformation preserves the comultiplication structure constants.
- Multiplication structure constants are governed by quantum 3j symbols.
- Links between preferred deformations and Schur-Weyl duality are established.

## Abstract

We discuss the deformed function algebra of a simply connected reductive Lie group G over the complex numbers using a basis consisting of matrix elements of finite dimensional representations. This leads to a preferred deformation, meaning one where the structure constants of comultiplication are unchanged. The structure constants of multiplication are controlled by quantum 3j symbols. We then discuss connections earlier work on preferred deformations that involved Schur-Weyl duality.