Clifford Algebras

• Introduction
• Clifford Algebras
  • CliffordAlgebra(Q): AlgMatElt → AlgClff, ModTupFld, Map
  • CliffordAlgebra(V): ModTupFld → AlgClff, Map
  • Example: Quaternions as a Clifford Algebra
  • Print Names for Generators
    • AssignNames(~C, S): AlgClff, [ MonStgElt ]
    • Name(C, i): AlgClff, RngIntElt → AlgClffElt
    • AsPolynomial(v): AlgClffElt
    • Example: Naming Clifford Algebra Generators
    • Example: Changing Print Names of Generators
  • Elements of a Clifford Algebra
    • elt< C | r₁, r₂, ..., rₘ >: AlgClff, RngElt, RngElt, ..., RngElt → AlgClffElt
    • C ! L: AlgClff, SeqEnum[RngElt] → AlgGenElt
    • SeqToClifford(C, ss): AlgClff, SeqEnum → AlgClffElt
    • SeqFromClifford(v): AlgClffElt → SeqEnum
    • Example: Monomial Representation of an Element
    • BasisProduct(C, i, j): AlgClff, RngIntElt, RngIntElt → AlgGenElt
    • BasisProduct(C, L): AlgClff, SeqEnum → AlgGenElt
    • BasisElement(C, S): AlgClff, SetEnum → AlgClffElt
• The Main Involutions
  • MainInvolution(C): AlgClff → Map
  • MainAntiautomorphism(C): AlgClff → Map
• Clifford Algebra Structure
  • HomogeneousComponent(v,k): AlgClffElt, RngIntElt → AlgClffElt
  • Example: Homogeneous Components of an Element
  • EvenSubalgebra(C : parameters): AlgClff → AlgAss, Map
  • Example: Quaternions as an Even Subalgebra
  • Centre(C): AlgClff → AlgAss, Map
  • Center(C): AlgClff → AlgAss, Map
  • Example: Even Dimension with Maximal Witt Index
  • Example: Even Dimension with Non-Maximal Witt Index
  • Example: Clifford Algebra in Odd Dimension
• Vector and Spin Representations
  • The Clifford Group
    • VectorAction(g): AlgClffElt → AlgMatElt
    • Example: Reflections from the Clifford Group
  • Siegel Transformations and Spin Groups
    • Example: Siegel Transformations
    • Example: Spin Preimages of Standard Generators
  • Spin Representations
    • ActionMatrix(S, s): AlgAss, AlgAssElt → AlgMatElt
    • Example: Minimal Right Ideals of the Even Subalgebra
    • Example: Constructing a Spin Representation
• Bibliography