Computational searches for fractional Calabi-Yau algebras

Introduction

This repository was created whilst working on my 2023 LMS URB project: Computational searches for fractional Calabi-Yau algebras, supervised by Dr Joseph Grant (UEA).

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There are two collections of code (written in Python and GAP) contained in py_src and gap_src respectively. Due to speed issues, the GAP code was eventually abandoned in favour of Python. As a result, the focal point of this repository is the Python code. The GAP code will not be documented nor explained, but is included for completeness.

Mathematical Motivation

Given $n \geq 3$, we can construct the linear quiver $Q_n$. This is a quiver of the form $$1 \rightarrow 2 \rightarrow \ldots \rightarrow n-1 \rightarrow n,$$ with the vertices naturally labelled by elements of $\mathbb{N}$. By then choosing a field $k$, we can use $Q_n$ to form the path algebra $kQ_n$, on which we can introduce relations. A relation is a non-zero path (of length $\geq 2$) which we define to be zero. This involves forming the quotient algebra $kQ_n / I$, where $I$ is the ideal generated by the relations.

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It is known that, if $I = ()$, then $kQ_n/I$ is fractional Calabi-Yau (fCY) for any $n$. However, less is known when $I$ is non-zero. This repository's major motivation was to produce code that allows for testing whether $kQ_n/I$ is fCY.

This is implemented in two different ways: (1) by testing whether the matrix obtained from the projective resolution of the injectives has finite order, and (2) by repeatedly applying the Serre functor until it is isomorphic to a shift, hence obtaining the Calabi-Yau dimension of the algebra.

Dependencies

This code has been written and tested for Python 3.10.12, though probably works on other versions.
NumPy is the only required library.

Getting Started

The best place to get started is by following the interactive tutorial tutorial.ipynb (requires Jupyter) in the py_src directory.