ExteriorExtensions -- Builds a \(\mathbb{Z}_m\) -graded algebra that equivariantly extends a Lie algebra

Description

ExteriorExtensions Builds a \(\mathbb{Z}_m\) -graded algebra that equivariantly extends a Lie algebra \( \mathfrak{sl}_n\) via the non-zero graded piece of the exterior algebra by defining the bracket products. Constructs matrix representations of adjoint operators. Computes ranks of blocks coming from the grading. Computing matrix representations of adjoint operators and computing ranks of their blocks. The user can also call for the Killing matrix of the algebra.

Author

Luke Oeding <oeding@auburn.edu>

Certification

Version 1.0 of this package was accepted for publication in volume 15 of Journal of Software for Algebra and Geometry on 2025-07-16, in the article ExteriorExtensions: a package for Macaulay2 (DOI: 10.2140/jsag.2025.15.69). That version can be obtained from the journal.

Version

This documentation describes version 1.0 of ExteriorExtensions, released April 23, 2025.

Citation

If you have used this package in your research, please cite it as follows:

@misc{ExteriorExtensionsSource,
  title = {{ExteriorExtensions: Builds an algebra extending an exterior algebra. Version~1.0}},
  author = {Luke Oeding},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

@article{ExteriorExtensionsArticle,
  title = {{ExteriorExtensions: a package for \emph{Macaulay2}}},
  author = {Luke Oeding},
  journal = {Journal of Software for Algebra and Geometry},
  volume = {15},
  year = {2025},
}

Exports

Types
- ExteriorExtension -- A new algebra constructed from the direct sum of a Lie algebra and a module
Functions and commands
- exteriorExtension -- Make an extension of an exterior algebra
- getDiagonal -- Extract the diagonal of a matrix
- KillingMatrix -- Constructs the Killing matrix of the ExteriorExtension
- makeTraceless -- Project matrix to the traceless matrices.
- structureTensor -- Constructs the structureTensor of the ExteriorExtension
Methods
- exteriorExtension(ZZ,ZZ) -- see exteriorExtension -- Make an extension of an exterior algebra
- exteriorExtension(ZZ,ZZ,Ring) -- see exteriorExtension -- Make an extension of an exterior algebra
- exteriorExtension(ZZ,ZZ,Symbol) -- see exteriorExtension -- Make an extension of an exterior algebra
- exteriorExtension(ZZ,ZZ,Symbol,Ring) -- see exteriorExtension -- Make an extension of an exterior algebra
- KillingMatrix(ExteriorExtension) (missing documentation)
- structureTensor(ExteriorExtension) (missing documentation)
Symbols
- ad -- Constructs the standard matrix representation of the adjoint operator
- appendage -- a placeholder for the object (exterior algebra) added on to the Lie algebra by exteriorExtension
- bases -- gives a list of the bases of the different graded pieces of the ExteriorExtension
- blockPowerRanks -- Computes a list of ranks of blocks for each power of a matrix representation of an adjoint operator for ExteriorExtensions
- bracket -- Compute products in an ExteriorExtension
- findGrade -- Find which graded piece of the algebra an element is in
- getBlock -- Extract a block of the matrix representation of an adjoint operator for ExteriorExtensions
- HodgeStar -- The Hodge star in an exterior algebra
- LieAlg2Matrix -- Converts from the LieAlgebra to Matrix form (with the standard basis)
- LieAlgebra -- a placeholder for the base Lie algebra
- matrix2LieAlg -- Converts from Matrix form to the LieAlgebra (with the standard basis)
- powerRanks -- Compute the ranks of powers of a matrix
- powerTraces -- Compute the traces of powers of a matrix
- prettyBlockPowerRanks -- same as blockPowerRanks, but prettier

For the programmer

The object ExteriorExtensions is a package, defined in ExteriorExtensions.m2.

The source of this document is in ExteriorExtensions.m2:288:0.