# Quadratic $d$-numbers

## TL;DR

This paper classifies quadratic $d$-numbers, explores their properties in quadratic fields, and introduces weakly quadratic fusion categories, linking algebraic integers to fusion category dimensions.

## Contribution

It provides a constructive classification of quadratic $d$-numbers and characterizes real quadratic fields with units of norm -1, introducing weakly quadratic fusion categories.

## Key findings

- Discreteness of maximal Galois conjugates set in real quadratic fields.
- Characterization of quadratic fields with units of norm -1.
- Systematic study framework for weakly quadratic fusion categories.

## Abstract

Here we constructively classify quadratic $d$-numbers: algebraic integers in quadratic number fields generating Galois-invariant ideals. We prove the subset thereof maximal among their Galois conjugates in absolute value is discrete in $\mathbb{R}$. Our classification provides a characterization of those real quadratic fields containing a unit of norm -1 which is known to be equivalent to the existence of solutions to the negative Pell equation. The notion of a weakly quadratic fusion category is introduced whose Frobenius-Perron dimension necessarily lies in this discrete set. Factorization, divisibility, and boundedness results are proven for quadratic $d$-numbers allowing a systematic study of weakly quadratic fusion categories which constitute essentially all known examples of fusion categories having no known connection to classical representation theory.