ExteriorExtensions: A package for Macaulay2

TL;DR

This paper introduces a Macaulay2 package called ExteriorExtensions that implements algebraic constructions related to Lie algebras and modules, enabling computations of brackets, Killing forms, and matrix representations.

Contribution

It provides a practical implementation of a graded algebra structure on Lie algebras and modules, based on a recent theoretical construction, facilitating computational experiments.

Findings

Implemented brackets, Killing form, and matrix representations in Macaulay2.

Enabled computation of ranks of blocks of adjoint operator powers.

Facilitated algebraic structure analysis for Lie algebras and modules.

Abstract

We explain a Macaulay2 implementation of a construction, which appeared in [Holweck-Oeding arXiv:2206.13662], of a graded algebra structure on the direct sum of a Lie algebra $\mathfrak{g}$ (typically $\mathfrak{sl}_n$) and a $\mathfrak{g}$-module (typically a subspace of an exterior algebra $\bigwedge^{\bullet}\mathbb{C}^n$). We implement brackets, a Killing form, matrix representations of adjoint operators and ranks of blocks of their powers.