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InternalModules

Modules over internal algebras

2024.12-01

6 December 2024

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Fabian Zickgraf
Email: fabian.zickgraf@uni-siegen.de
Homepage: https://algebra.mathematik.uni-siegen.de/zickgraf
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1.1 Examples for the internal symmetric algebra

  1.1-1 The tensor category of finite dimensional GF(2)-vector spaces
  1.1-2 The tensor category of finite dimensional Q-vector spaces
  1.1-3 The tensor category of finite dimensional Q-representations of the symmetric group Sym(4).
  1.1-4 The tensor category of finite dimensional Z-graded Q-representations of the symmetric group Sym(4).
  1.1-5 Futher computations in the symmetric algebra

1.2 Examples for the internal exterior algebra

1.2-1 The tensor category of finite dimensional Q-vector spaces

4 Internal module elements

4.1-1 CoefficientsVector

4.2 GAP Categories

  4.2-1 IsElementInInternalModule
  4.2-2 IsElementInInternalLeftModule
  4.2-3 IsElementInInternalRightModule

5 Internal exterior or symmetric algebra

5.1-1 ExteriorOrSymmetricAlgebraMultiplication

6 Internal symmetric algebra

  6.1-1 SymmetricAlgebraAsZFunction
  6.1-2 SymmetricAlgebraMultiplicationListList
  6.1-3 SymmetricAlgebra
  6.1-4 SymmetricAlgebraMultiplicationMorphism
  6.1-5 SymmetricAlgebraAsLeftModule
  6.1-6 SymmetricAlgebraAsRightModule

6.2 GAP Categories

6.2-1 IsInternalSymmetricAlgebra

7 Internal exterior algebra

  7.1-1 ExteriorAlgebraAsZFunction
  7.1-2 ExteriorAlgebraMultiplicationListList
  7.1-3 ExteriorAlgebra
  7.1-4 ExteriorAlgebraMultiplicationMorphism
  7.1-5 ExteriorAlgebraAsLeftModule
  7.1-6 ExteriorAlgebraAsRightModule

7.2 GAP Categories

7.2-1 IsInternalExteriorAlgebra

8 Modules over the internal symmetric algebra

8.1 Constructors

8.1-1 CategoryOfLeftSModules
8.1-2 CategoryOfRightSModules

8.2 GAP Categories

  8.2-1 IsCategoryOfSModules
  8.2-2 IsCategoryOfLeftSModules
  8.2-3 IsCategoryOfRightSModules
  8.2-4 IsSModuleCell
  8.2-5 IsSModule
  8.2-6 IsLeftSModule
  8.2-7 IsRightSModule
  8.2-8 IsSModuleMorphism
  8.2-9 IsLeftSModuleMorphism
  8.2-10 IsRightSModuleMorphism

9 Modules over the internal exterior algebra

9.1 GAP Categories

  9.1-1 IsCategoryOfEModules
  9.1-2 IsCategoryOfLeftEModules
  9.1-3 IsCategoryOfRightEModules
  9.1-4 IsEModuleCell
  9.1-5 IsEModule
  9.1-6 IsLeftEModule
  9.1-7 IsRightEModule
  9.1-8 IsEModuleMorphism
  9.1-9 IsLeftEModuleMorphism
  9.1-10 IsRightEModuleMorphism

9.2 Constructors

9.2-1 CategoryOfLeftEModules
9.2-2 CategoryOfRightEModules

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