Algebras

• Catalog of Algebras
• Quantum Groups

Free associative algebras and quotients

• Free algebras
  • FreeAlgebraFactory
  • FreeAlgebra_generic
  • PBWBasisOfFreeAlgebra
  • is_FreeAlgebra()
• Free algebra elements
  • FreeAlgebraElement
• Free associative unital algebras, implemented via Singular’s letterplace rings
  • FreeAlgebra_letterplace
  • FreeAlgebra_letterplace_libsingular
  • freeAlgebra()
• Weighted homogeneous elements of free algebras, in letterplace implementation
  • FreeAlgebraElement_letterplace
  • poly_reduce()
• Homogeneous ideals of free algebras
  • LetterplaceIdeal
  • poly_reduce()
  • singular_twostd()
• Finite dimensional free algebra quotients
  • FreeAlgebraQuotient
  • hamilton_quatalg()
• Free algebra quotient elements
  • FreeAlgebraQuotientElement
  • is_FreeAlgebraQuotientElement()
• Tensor Algebras
  • BaseRingLift
  • TensorAlgebra
  • TensorAlgebraFunctor

Finite dimensional algebras

• Finite-Dimensional Algebras
  • FiniteDimensionalAlgebra
• Elements of Finite Algebras
  • FiniteDimensionalAlgebraElement
  • unpickle_FiniteDimensionalAlgebraElement()
• Ideals of Finite Algebras
  • FiniteDimensionalAlgebraIdeal
• Morphisms Between Finite Algebras
  • FiniteDimensionalAlgebraHomset
  • FiniteDimensionalAlgebraMorphism

Named associative algebras

• Affine nilTemperley Lieb Algebra of type A
  • AffineNilTemperleyLiebTypeA
• Askey-Wilson Algebras
  • AlgebraMorphism
  • AskeyWilsonAlgebra
• Diagram and Partition Algebras
  • AbstractPartitionDiagram
  • AbstractPartitionDiagrams
  • BrauerAlgebra
  • BrauerDiagram
  • BrauerDiagrams
  • DiagramAlgebra
  • DiagramBasis
  • IdealDiagram
  • IdealDiagrams
  • OrbitBasis
  • PartitionAlgebra
  • PartitionDiagram
  • PartitionDiagrams
  • PlanarAlgebra
  • PlanarDiagram
  • PlanarDiagrams
  • PropagatingIdeal
  • SubPartitionAlgebra
  • TL_diagram_ascii_art()
  • TemperleyLiebAlgebra
  • TemperleyLiebDiagram
  • TemperleyLiebDiagrams
  • UnitDiagramMixin
  • brauer_diagrams()
  • diagram_latex()
  • ideal_diagrams()
  • identity_set_partition()
  • is_planar()
  • pair_to_graph()
  • partition_diagrams()
  • planar_diagrams()
  • planar_partitions_rec()
  • propagating_number()
  • temperley_lieb_diagrams()
  • to_Brauer_partition()
  • to_graph()
  • to_set_partition()
• Clifford Algebras
  • CliffordAlgebra
  • CliffordAlgebraIndices
  • ExteriorAlgebra
  • ExteriorAlgebraBoundary
  • ExteriorAlgebraCoboundary
  • ExteriorAlgebraDifferential
  • ExteriorAlgebraIdeal
• Cluster algebras
  • ClusterAlgebra
  • ClusterAlgebraElement
  • ClusterAlgebraSeed
  • PrincipalClusterAlgebraElement
• Descent Algebras
  • DescentAlgebra
  • DescentAlgebraBases
• Fusion Rings
  • Fusion Rings
  • The F-Matrix of a Fusion Ring
  • F-Matrix Backend
• Hall Algebras
  • HallAlgebra
  • HallAlgebraMonomials
  • transpose_cmp()
• Incidence Algebras
  • IncidenceAlgebra
  • ReducedIncidenceAlgebra
• Group algebras
  • GroupAlgebra()
  • GroupAlgebra_class
• Grossman-Larson Hopf Algebras
  • GrossmanLarsonAlgebra
• Möbius Algebras
  • BasisAbstract
  • MoebiusAlgebra
  • MoebiusAlgebraBases
  • QuantumMoebiusAlgebra
• Orlik-Terao Algebras
  • OrlikTeraoAlgebra
  • OrlikTeraoInvariantAlgebra
• Orlik-Solomon Algebras
  • OrlikSolomonAlgebra
  • OrlikSolomonInvariantAlgebra
• Partition/Diagram Algebras
  • PartitionAlgebraElement_ak
  • PartitionAlgebraElement_bk
  • PartitionAlgebraElement_generic
  • PartitionAlgebraElement_pk
  • PartitionAlgebraElement_prk
  • PartitionAlgebraElement_rk
  • PartitionAlgebraElement_sk
  • PartitionAlgebraElement_tk
  • PartitionAlgebra_ak
  • PartitionAlgebra_bk
  • PartitionAlgebra_generic
  • PartitionAlgebra_pk
  • PartitionAlgebra_prk
  • PartitionAlgebra_rk
  • PartitionAlgebra_sk
  • PartitionAlgebra_tk
  • SetPartitionsAk()
  • SetPartitionsAk_k
  • SetPartitionsAkhalf_k
  • SetPartitionsBk()
  • SetPartitionsBk_k
  • SetPartitionsBkhalf_k
  • SetPartitionsIk()
  • SetPartitionsIk_k
  • SetPartitionsIkhalf_k
  • SetPartitionsPRk()
  • SetPartitionsPRk_k
  • SetPartitionsPRkhalf_k
  • SetPartitionsPk()
  • SetPartitionsPk_k
  • SetPartitionsPkhalf_k
  • SetPartitionsRk()
  • SetPartitionsRk_k
  • SetPartitionsRkhalf_k
  • SetPartitionsSk()
  • SetPartitionsSk_k
  • SetPartitionsSkhalf_k
  • SetPartitionsTk()
  • SetPartitionsTk_k
  • SetPartitionsTkhalf_k
  • SetPartitionsXkElement
  • identity()
  • is_planar()
  • pair_to_graph()
  • propagating_number()
  • set_partition_composition()
  • to_graph()
  • to_set_partition()
• Quantum Clifford Algebras
  • QuantumCliffordAlgebra
  • QuantumCliffordAlgebraGeneric
  • QuantumCliffordAlgebraRootUnity
• Quantum Groups Using GAP’s QuaGroup Package
  • CrystalGraphVertex
  • HighestWeightModule
  • HighestWeightSubmodule
  • LowerHalfQuantumGroup
  • QuaGroupModuleElement
  • QuaGroupRepresentationElement
  • QuantumGroup
  • QuantumGroupHomset
  • QuantumGroupModule
  • QuantumGroupMorphism
  • TensorProductOfHighestWeightModules
  • projection_lower_half()
• Quantum Matrix Coordinate Algebras
  • QuantumGL
  • QuantumMatrixCoordinateAlgebra
  • QuantumMatrixCoordinateAlgebra_abstract
• Quaternion Algebras
  • QuaternionAlgebraFactory
  • QuaternionAlgebra_ab
  • QuaternionAlgebra_abstract
  • QuaternionFractionalIdeal
  • QuaternionFractionalIdeal_rational
  • QuaternionOrder
  • basis_for_quaternion_lattice()
  • intersection_of_row_modules_over_ZZ()
  • is_QuaternionAlgebra()
  • maxord_solve_aux_eq()
  • normalize_basis_at_p()
  • unpickle_QuaternionAlgebra_v0()
• Rational Cherednik Algebras
  • RationalCherednikAlgebra
• Schur algebras for \(GL_n\)
  • GL_irreducible_character()
  • SchurAlgebra
  • SchurTensorModule
  • schur_representative_from_index()
  • schur_representative_indices()
• The Steenrod algebra
  • AA()
  • Sq()
  • SteenrodAlgebra()
  • SteenrodAlgebra_generic
  • SteenrodAlgebra_mod_two
• Steenrod algebra bases
  • arnonC_basis()
  • atomic_basis()
  • atomic_basis_odd()
  • convert_from_milnor_matrix()
  • convert_to_milnor_matrix()
  • milnor_basis()
  • restricted_partitions()
  • serre_cartan_basis()
  • steenrod_algebra_basis()
  • steenrod_basis_error_check()
  • xi_degrees()
• Miscellaneous functions for the Steenrod algebra and its elements
  • arnonA_long_mono_to_string()
  • arnonA_mono_to_string()
  • comm_long_mono_to_string()
  • comm_mono_to_string()
  • convert_perm()
  • get_basis_name()
  • is_valid_profile()
  • milnor_mono_to_string()
  • normalize_profile()
  • pst_mono_to_string()
  • serre_cartan_mono_to_string()
  • wall_long_mono_to_string()
  • wall_mono_to_string()
  • wood_mono_to_string()
• Multiplication for elements of the Steenrod algebra
  • adem()
  • binomial_mod2()
  • binomial_modp()
  • make_mono_admissible()
  • milnor_multiplication()
  • milnor_multiplication_odd()
  • multinomial()
  • multinomial_odd()
• Weyl Algebras
  • DifferentialWeylAlgebra
  • DifferentialWeylAlgebraAction
  • DifferentialWeylAlgebraElement
  • repr_factored()
  • repr_from_monomials()
• Yangians
  • GradedYangianBase
  • GradedYangianLoop
  • GradedYangianNatural
  • Yangian
  • YangianLevel

Hecke algebras

• Ariki-Koike Algebras
  • ArikiKoikeAlgebra
• Iwahori-Hecke Algebras
  • IwahoriHeckeAlgebra
  • IwahoriHeckeAlgebra_nonstandard
  • index_cmp()
  • normalized_laurent_polynomial()
• Nil-Coxeter Algebra
  • NilCoxeterAlgebra
• Yokonuma-Hecke Algebras
  • YokonumaHeckeAlgebra
• Cubic Hecke Algebras
  • Cubic Hecke Algebras
  • Cubic Hecke Base Rings
  • Cubic Hecke matrix representations

Graded algebras

• Finite dimensional graded commutative algebras
  • FiniteGCAlgebra
• Commutative Differential Graded Algebras
  • CohomologyClass
  • Differential
  • DifferentialGCAlgebra
  • DifferentialGCAlgebra_multigraded
  • Differential_multigraded
  • GCAlgebra
  • GCAlgebraHomset
  • GCAlgebraMorphism
  • GCAlgebra_multigraded
  • GradedCommutativeAlgebra()
  • exterior_algebra_basis()
  • sorting_keys()
  • total_degree()

Various associative algebras

• Associated Graded Algebras To Filtered Algebras
  • AssociatedGradedAlgebra
• Cellular Basis
  • CellularBasis
• Q-Systems
  • QSystem
  • is_tamely_laced()
• \(q\)-Commuting Polynomials
  • qCommutingPolynomials
• Splitting Algebras
  • SplittingAlgebra
  • SplittingAlgebraElement
  • solve_with_extension()

Non-associative algebras

• Lie Algebras
  • Abelian Lie Algebras
  • Affine Lie Algebras
  • The Baker-Campbell-Hausdorff formula
  • Classical Lie Algebras
  • Examples of Lie Algebras
  • Free Lie Algebras
  • Heisenberg Algebras
  • Lie Algebras
  • Lie Algebra Elements
  • Homomorphisms of Lie Algebras
  • Nilpotent Lie algebras
  • Onsager Algebra
  • The Poincare-Birkhoff-Witt Basis For A Universal Enveloping Algebra
  • Quotients of Lie algebras
  • Rank Two Heisenberg-Virasoro Algebras
  • Lie Algebras Given By Structure Coefficients
  • Subalgebras and ideals of Lie algebras
  • Symplectic Derivation Lie Algebras
  • Verma Modules
  • Virasoro Algebra and Related Lie Algebras
• Lie Conformal Algebras
  • The main classes to work with Lie conformal algebras
  • Implemented examples of Lie Conformal Algebras
  • See also
• Jordan Algebras
  • JordanAlgebra
  • JordanAlgebraSymmetricBilinear
  • SpecialJordanAlgebra
• Free Dendriform Algebras
  • DendriformFunctor
  • FreeDendriformAlgebra
• Free Pre-Lie Algebras
  • FreePreLieAlgebra
  • PreLieFunctor
• Shuffle algebras
  • DualPBWBasis
  • ShuffleAlgebra
• Free Zinbiel Algebras
  • FreeZinbielAlgebra
  • ZinbielFunctor

Indices and Tables

• Index

• Module Index

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