# Clifford Algebras

- [Introduction](introduction.md)

- [Clifford Algebras](creation.md)

  - [`CliffordAlgebra(Q): AlgMatElt → AlgClff, ModTupFld, Map`](creation.md#function-cliffordalgebra-algmatelt)

  - [`CliffordAlgebra(V): ModTupFld → AlgClff, Map`](creation.md#function-cliffordalgebra-modtupfld)

  - [`Example: Cliff Quat Ex`](creation.md#example-ex-22ab1c)

  - [Print Names for Generators](creation.md#print-names-for-generators)

    - [`AssignNames(~C, S): AlgClff, [ MonStgElt ])`](creation.md#function-assignnames-algclff-monstgelt-ref)

    - [`Name(C, i): AlgClff, RngIntElt → AlgClffElt`](creation.md#function-name-algclff-rngintelt)

    - [`AsPolynomial(v): AlgClffElt`](creation.md#function-aspolynomial-algclffelt)

    - [`Example: Clifford Gen`](creation.md#example-ex-dfadc4)

    - [`Example: Clifford Names`](creation.md#example-ex-436f79)

  - [Elements of a Clifford Algebra](creation.md#elements-of-a-clifford-algebra)

    - [`elt< C | r₁, r₂, ..., rₘ >: AlgClff, RngElt, RngElt, ..., RngElt → AlgClffElt`](creation.md#constructor-constructor-elt-algclff-rngelt-rngelt-rngelt-algclffelt)

    - [`C ! L: AlgClff, SeqEnum[RngElt] → AlgGenElt`](creation.md#operation-op-algclff-seqenum-rngelt)

    - [`SeqToClifford(C, ss): AlgClff, SeqEnum → AlgClffElt`](creation.md#function-seqtoclifford-algclff-seqenum)

    - [`SeqFromClifford(v): AlgClffElt → SeqEnum`](creation.md#function-seqfromclifford-algclffelt)

    - [`Example: Eltfromseq`](creation.md#example-ex-3fca24)

    - [`BasisProduct(C, i, j): AlgClff, RngIntElt, RngIntElt → AlgGenElt`](creation.md#function-basisproduct-algclff-rngintelt-rngintelt)

    - [`BasisProduct(C, L): AlgClff, SeqEnum → AlgGenElt`](creation.md#function-basisproduct-algclff-seqenum)

    - [`BasisElement(C, S): AlgClff, SetEnum → AlgClffElt`](creation.md#function-basiselement-algclff-setenum)

- [The Main Involutions](involutions.md)

  - [`MainInvolution(C): AlgClff → Map`](involutions.md#function-maininvolution-algclff)

  - [`MainAntiautomorphism(C): AlgClff → Map`](involutions.md#function-mainantiautomorphism-algclff)

- [Clifford Algebra Structure](operations.md)

  - [`HomogeneousComponent(v,k): AlgClffElt, RngIntElt → AlgClffElt`](operations.md#function-homogeneouscomponent-algclffelt-rngintelt)

  - [`Example: Hom Comp`](operations.md#example-ex-40ad8c)

  - [`EvenSubalgebra(C : parameters): AlgClff → AlgAss, Map`](operations.md#function-evensubalgebra-algclff)

  - [`Example: Cliff Gen Quat Ex`](operations.md#example-ex-521a9d)

  - [`Centre(C): AlgClff → AlgAss, Map`](operations.md#function-centre-algclff)

  - [`Center(C): AlgClff → AlgAss, Map`](operations.md#function-center-algclff)

  - [`Example: Even Dim Plus`](operations.md#example-ex-d39588)

  - [`Example: Even Dim Minus`](operations.md#example-ex-04b7d8)

  - [`Example: Odd Dim`](operations.md#example-ex-a8b683)

- [Vector and Spin Representations](spin.md)

  - [The Clifford Group](spin.md#the-clifford-group)

    - [`VectorAction(g): AlgClffElt → AlgMatElt`](spin.md#function-vectoraction-algclffelt)

    - [`Example: Clff Ref`](spin.md#example-ex-6e1ea1)

  - [Siegel Transformations and Spin Groups](spin.md#siegel-transformations-and-spin-groups)

    - [`Example: siegel`](spin.md#example-ex-c93d8c)

    - [`Example: vecrep`](spin.md#example-ex-1c9644)

  - [Spin Representations](spin.md#spin-representations)

    - [`ActionMatrix(S, s): AlgAss, AlgAssElt → AlgMatElt`](spin.md#function-actionmatrix-algass-algasselt)

    - [`Example: ideals`](spin.md#example-ex-b3def3)

    - [`Example: spinrep`](spin.md#example-ex-d57a8d)