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Oscar.jl
  • Welcome to OSCAR
    • Architecture
    • Notes for users of other computer algebra systems
    • Printing Options
    • Frequently Asked Questions
    • Saving and loading files
    • Complex Algorithms
    • Introduction
    • Basics
    • Subgroups
    • Quotients
    • Products of groups
    • Permutation groups
    • Finitely presented groups
    • Polycyclic groups
    • Matrix groups
    • Group actions
    • Group homomorphisms
    • Groups of automorphisms
    • Group libraries
      • Introduction
      • Elements
      • Morphisms
      • Structural Computations
    • Group characters
    • Group recognition
    • Tables of Marks
    • Introduction
    • Ring functionality
    • Integers
      • Univariate polynomial functionality
      • Sparse distributed multivariate polynomials
      • Power series
      • Generic Puiseux series
      • Power series and Laurent series
      • Puiseux series
    • Introduction
    • Field functionality
    • Rationals
    • Factored Elements
    • Class Field Theory
    • Generic fraction fields
      • Padics
      • Qadics
    • Finite fields
    • Algebraic numbers
    • Algebraic closure of finite prime fields
    • Introduction
    • Matrix functionality
    • Matrix Spaces
    • Linear solving
    • Generic matrix algebras
    • Sparse linear algebra
      • Finitely presented modules
      • Free Modules and Vector Spaces
      • Submodules
      • Quotient modules
      • Direct Sums
      • Module Homomorphisms
      • Introduction
      • Spaces
      • Lattices
      • Genera for hermitian lattices
      • Integer Lattices
      • Genera of Integer Lattices
      • Discriminant Groups
      • Vinberg's algorithm
    • Matrix implementation
    • Introduction
      • Introduction
      • Number field operations
      • Element operations
      • Internals
      • Introduction
      • Orders
      • Elements
      • Ideals
      • Fractional ideals
      • Introduction
      • Basics
      • Elliptic curves over finite fields
    • Abelian closure of the rationals
    • Galois Theory
    • Introduction
      • Introduction
      • Constructions
      • Polyhedron and polymake's Polytope
      • Auxiliary functions
    • Cones
    • Polyhedral Fans
    • Polyhedral Complexes
    • Subdivisions of Points
      • Linear Programs
      • Mixed Integer Linear Programs
    • Introduction
    • Creating Multivariate Rings
    • Ideals in Multivariate Rings
    • Affine Algebras and Their Ideals
    • Localized Rings and Their Ideals
      • Introduction
      • Free Modules
      • Subquotients
      • Ideals and Quotient Rings as Modules
      • Operations on Modules
      • Operations on Module Maps
      • Chain and Cochain Complexes
    • Homological Algebra
      • Monomial Orderings
      • Gröbner/Standard Bases Over Fields
      • Gröbner/Standard Bases Over $\mathbb Z$
      • Binomial Primary Decomposition
      • A Framework for Localizing Rings
      • Localizations of modules over computable rings
    • Introduction
    • Invariants of Finite Groups
    • Invariants of Tori
    • Invariants of Linearly Reductive Groups
    • Introduction
      • General schemes
      • Affine schemes
      • Morphisms of affine schemes
      • Rational Points on Affine Schemes
      • Covered schemes
      • Coverings
      • Morphisms of covered schemes
      • Projective schemes
      • Morphisms of projective schemes
      • Rational Points on Projective Schemes
      • Sheaves on covered schemes
      • Cycles and divisors
      • Affine Algebraic Sets
      • Projective Algebraic Sets
      • Affine Varieties
      • Projective Varieties
      • Affine plane curves
      • Projective Curves
      • Projective Plane Curves
      • Rational Parametrizations of Rational Plane Curves
      • Automorphism Groups of K3 surfaces
      • Borcherds' method for Enriques surfaces
      • Elliptic Surfaces
      • Adjunction Process for Surfaces
      • Rational Parametrization of Rational Surfaces
      • Nongeneral Type Surfaces in $\mathbb P^4$
      • Classifier/identifier specifically for du Val singularities
      • Sheaves on Projective Space
      • Introduction
      • Normal Toric Varieties
      • Cyclic Quotient Singularities
      • Toric Divisors
      • Toric Divisor Classes
      • Toric Line Bundles
      • Line bundle cohomology with cohomCalg
      • Cohomology Classes
      • Subvarieties
      • The Chow ring
      • ToricMorphisms
      • Toric Schemes
      • Toric Ideal Sheaves (Experimental)
      • Toric Blowups (Experimental)
      • Some Special Ideals
      • Architecture of affine schemes
    • Introduction
    • Tropical semirings, matrices, and polynomials
    • Tropical semiring maps
    • Tropical varieties
    • Tropical hypersurfaces
    • Tropical curves
    • Tropical linear spaces
    • Tropical Groebner theory
    • Tropicalization of polynomial ideals
    • Positive tropicalizations of linear ideals
    • Introduction
      • Introduction
      • Creating PBW-Algebras
      • Ideals in PBW-algebras
      • GR-Algebras: Quotients of PBW-Algebras
    • Free Associative Algebras
    • Introduction
      • Contact
    • Cartan matrices
    • Root systems
    • Weight lattices
    • Weyl groups
    • Graphs
    • Matroids
    • Simplicial Complexes
    • Phylogenetic Trees
      • Partitions
      • Multipartitions
      • Tableaux
      • Schur polynomials
      • Compositions
    • Introduction
    • GAP's SLPs
    • AbstractAlgebra's polynomial interface
  • References
  • Index
    • Introduction for new developers
    • Developer Style Guide
    • Documenting Code
    • Printing Details
    • Debugging Code
    • Caching Parent Objects
    • Serialization
    • Design Decisions
    • GAP Integration
    • Release management
    • Updating CHANGELOG.md
      • AbstractCollection
      • SubObjectIterator
    • Writing parallel methods
    • Adding new projects to experimental
      • Introduction
      • Graphical Models
      • Gaussian Graphical Models
      • Conditional independence statements
      • Discrete random variables
      • Algebraic Phylogenetics
      • Double complexes – the user's interface
      • Advice for the programmer
      • Introduction
      • An example documentation
      • Usage
      • Special ideals used for benchmarking
      • Introduction
      • Construction and basic functionality
      • Cox rings
      • Matroid Realization Spaces
      • Introduction
      • Access to precomputed OD data
      • Criteria for computing orthogonal discriminants
      • Miscellaneous functions
      • Parallelization in Oscar
      • $q$-analogs
      • Quadratic forms and isometries
      • Quadratic spaces with isometry
      • Lattices with isometry
      • Enumeration of isometries
      • Nikulin's theory on primitive embeddings
      • -
      • Introduction
      • Lie algebras
      • Ideals and Lie subalgebras
      • Lie algebra homomorphisms
      • Lie algebra modules
      • Lie algebra module homomorphisms
      • Weyl groups (experimental features)
      • Introduction
      • Functions for a monomial basis of highest weight modules
      • Introduction
      • Exterior Shifting
      • Partial Shift Graph
      • Partitioned Permutations
      • Welcome to FTheoryTools
      • Functionality for all F-theory models
      • Weierstrass models
      • Global Tate models
      • Hypersurface models
      • Literature constructions
      • G4-Fluxes
      • Introduction
      • Abstract Varieties
      • Abstract Bundles
      • Abstract Variety Maps
      • Blow-Ups
      • Schubert Calculus
      • Localization and Bott's Formula
      • Illustrating Examples From Enumerative Geometry
Version
  • Lie Theory
  • Introduction
  • Introduction
GitHub

Introduction

The Lie theory part of OSCAR provides functionality for handling

  • Cartan matrices
  • Root systems
  • Weyl groups
  • Weight lattices
  • more content can be found under Experimental/LieAlgebras

It aims to provide the computational tools to work with the concepts defined in [Hum72] and [Bou02].

Contact

Please direct questions about this part of OSCAR to the following people:

  • Lars Göttgens
  • Max Horn.
  • Felix Röhrich

You can ask questions in the OSCAR Slack.

Alternatively, you can raise an issue on github.

« Free Associative AlgebrasCartan matrices »

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