Lie algebra gradings

This is a companion repository to the paper Gradings for nilpotent Lie algebras by Eero Hakavuori, Ville Kivioja, Terhi Moisala, and Francesca Tripaldi. The repository contains an implementation of the construction of stratifications, maximal gradings, and the enumeration of gradings for Lie algebras. It also includes a study of gradings for the nilpotent Lie algebra classifications of Cicalò-de Graaf-Schneider up to dimension 6 and Gong for dimension 7.

Usage

Prerequisites

A working installation of SageMath is needed. The code has been tested to work on version 9.0, released 2020-01-01.

Clone

Clone the repo with

$ git clone https://github.com/ehaka/lie-algebra-gradings

Test an example

Test any of the examples in the examples subfolder, for instance

$ cd lie-algebra-gradings
$ cd examples
$ sage second_heisenberg.sage

Run the automated doctests (optional)

Most of the core functionality has automated tests. These can be run with the call

sage -t dim7/isom_utilities.py lie_gradings/classification/*.py lie_gradings/gradings/*.py

The lie_gradings package

There is no separate installation, everything assumes that the package lie_gradings is found in sys.path. E.g. in examples/second_heisenberg.sage this is guaranteed with the preamble

import sys
import pathlib
path = pathlib.Path().absolute().parent
sys.path.append(str(path))

Here the assumption is that pathlib.Path().absolute() refers to the folder lie-algebra-gradings/examples, so the folder added to path is lie-algebra-gradings.

Note that sys.path when running Sage is different from $PATH.

Use with Sage and the Sage notebook

The easiest way to use the package within Sage is to directly launch Sage in the main folder of the repo

$ cd lie-algebra-gradings
$ sage

Similarly for the Sage notebook

$ cd lie-algebra-gradings
$ sage --notebook=jupyter

Then the functions in the lie_gradings package can be imported with

sage: from lie_gradings import *

Otherwise, the repo folder can be added to sys.path for the duration of a Sage session with sys.path.append('PATH-TO-LIE-ALGEBRA-GRADINGS-DIRECTORY') command.

Computing gradings for Lie algebras

The scripts in the examples subfolder demonstrate some of the basic functionality available. The basic structure of computations is the following.

First, import the contents of the lie_gradings package.

sage: from lie_gradings import *

Define a Lie algebra in Sage. The classifications of Cicalò-de Graaf-Schneider for dimensions 1-6 and of Gong for dimension 7 are implemented in lie_gradings/classification. For example the second Heisenberg Lie algebra is named L_{5,4} and can be constructed with coefficients in the field of algebraic numbers by

sage: L = d5.L5_4(QQbar)

The maximal grading of a Lie algebra can be computed using

sage: maximal_grading(L)

Some of the other main features are enumeration of torsion-free gradings by torsion_free_gradings(L) and computing a stratification if one exists by stratification(L).

The documentation of the various functions can be accessed with the syntax

sage: maximal_grading?

Data for nilpotent Lie algebras up to dimension 7

A listing of isomorphism classes of gradings for dimension up to 7 can be found in the dim7/data subfolder, along with the code used to construct the data in the dim7 subfolder. Up to dimension 6 the data is a complete listing of all possible torsion-free gradings based on the classification of nilpotent Lie algebras up to dimension 6 by Cicalò-de Graaf-Schneider. In dimension 7, the data is a complete listing apart from the one-parameter families containing an uncountable number of Lie algebras, based on the classification of nilpotent Lie algebras in dimension 7 by Gong.

The gradings can be explored in html format within the files under dim7/data/html and various text format overviews are available in dim7/data.

The precomputed gradings and isomorphism classes can be accessed directly as Sage objects through the various files in the subfolders of dim7/data/. See the data access example in the examples directory for a sample how to load these files into Sage for further computations.

Authors

• Eero Hakavuori
• Ville Kivioja
• Terhi Moisala
• Francesca Tripaldi

License

This project is licensed under the MIT License - see the LICENSE.txt file for details.