Quadratic $d$numbers
Abstract
Here we constructively classify quadratic $d$numbers: algebraic integers in quadratic number fields generating Galoisinvariant 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 FrobeniusPerron 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.09418
 Bibcode:
 2019arXiv190409418S
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Quantum Algebra
 EPrint:
 26 pages. Comments welcome