Fusion

Pure mathematics

Mathematics

Endomorphism

Algebra over a field

Philosophy

Artificial intelligence

Natural language processing

Computer science

Linguistics

SciPost Physics

Many applications of fusion categories, particularly in physics, require the associators or F <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>F</mml:mi></mml:math> -symbols to be known explicitly. Finding these matrices typically involves solving vast systems of coupled polynomial equations in large numbers of variables. In this work, we present an algorithm that allows associator data for some category with unknown associator to be computed from a Morita equivalent category with known data. Given a module category over the latter, we utilize the representation theory of a module tube category, built from the known data, to compute this unknown associator data. When the input category is unitary, we discuss how to ensure the obtained data is also unitary. We provide several worked examples to illustrate this algorithm. In addition, we include several Mathematica files showing how the algorithm can be used to compute the data for the Haagerup category \mathcal{H}_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mstyle mathvariant="script"><mml:mi>ℋ</mml:mi></mml:mstyle><mml:mn>1</mml:mn></mml:msub></mml:math> , whose data was previously unknown.

Computing associators of endomorphism fusion categories