# SU(2) and SU(1,1) Y-Maps in Loop Quantum Gravity

## TL;DR

This paper proves the convergence of $SU(2)$ and $SU(1,1)$ Y-Maps in Loop Quantum Gravity, establishing their mathematical validity and extending the framework to include $SU(1,1)$ structures.

## Contribution

It introduces the first proof of $SU(2)$ Y-Map convergence and defines a new $SU(1,1)$ Y-Map using simplicity constraints, demonstrating its convergence.

## Key findings

- Proved convergence of $SU(2)$ Y-Map.
- Defined and proved convergence of $SU(1,1)$ Y-Map.
- Extended Y-Map framework to $SU(1,1)$ in LQG.

## Abstract

In this paper we first provide the proof of $SU(2)$ Y-Map convergence. Then, by using $SU(1,1)$ LQG simplicity constraints we define $SU(1,1)$ Y-Map from infinitely differentiable with a compact support functions on $SU(1,1)$ to the functions (not necessarily square integrable) on $SL(2,C)$, and prove its convergence as well.