Algebras with Involution

• Introduction
• Algebras with Involution
  • Reflexive Forms
    • IsometryGroup(F : parameters): AlgMatElt → GrpMat
    • SimilarityGroup(F : parameters): AlgMatElt → GrpMat
  • Systems of Reflexive Forms
    • PGroupToForms(G): GrpPC → SeqEnum
    • PGroupToForms(G,p): GrpPC, RngIntElt → SeqEnum
    • PGroupToForms(G): GrpMat → SeqEnum
    • PGroupToForms(G,p): GrpMat, RngIntElt → SeqEnum
    • Example: Reflexive Forms from a p-Group
  • Basic Attributes of \(\ast\)-Algebras
    • IsStarAlgebra(A): AlgMat → BoolElt
    • IsStarAlgebra(A): AlgGrp → BoolElt
    • Star(A): AlgMat → Map
    • Star(A): AlgGrp → Map
  • Adjoint Algebras
    • AdjointAlgebra(S : parameters): SeqEnum → AlgMat
    • Example: Adjoint Algebra of a Pair of Forms
  • Group Algebras
    • StarOnGroupAlgebra(A): AlgGrp → Map
    • GroupAlgebraAsStarAlgebra(R, G): Rng, Grp → AlgGrp
    • Example: Group Algebra as a Star-Algebra
    • Example: Attaching an Involution to a Group Algebra
  • Simple \(\ast\)-Algebras
    • SimpleStarAlgebra(name, d, K): MonStgElt, RngIntElt, FldFin → AlgMat
    • Example: Simple Star-Algebra of Exchange Type
• Decompositions of \(\ast\)-Algebras
  • WedderburnDecomposition(A): AlgMat → AlgMat, AlgMat
  • WedderburnDecomposition(A): AlgGrp → AlgGrpSub, AlgGrpSub
  • TaftDecomposition(A): AlgMat → AlgMat, AlgMat
  • TaftDecomposition(A): AlgGrp → AlgGrpSub, AlgGrpSub
  • Example: Taft Decomposition of a Group Algebra
• Recognition of \(\ast\)-Algebras
  • Recognition of Simple \(\ast\)-Algebras
    • RecogniseClassicalSSA(A): AlgMat → BoolElt, AlgMat, Map, Map
    • RecogniseExchangeSSA(A): AlgMat → BoolElt, AlgMat, Map, Map
    • Example: Recognising a Simple Star-Algebra
  • Recognition of Arbitrary \(\ast\)-Algebras
    • RecogniseStarAlgebra(A): AlgMat → BoolElt
    • RecogniseStarAlgebra(A): AlgGrp → BoolElt
    • IsSimpleStarAlgebra(A): AlgMat → BoolElt
    • IsSimpleStarAlgebra(A): AlgGrp → BoolElt
    • SimpleParameters(A): AlgMat → SeqEnum
    • SimpleParameters(A): AlgGrp → SeqEnum
    • NormGroup(A): AlgMat → GrpMat
    • Example: Distinguishing Dihedral and Quaternion Group Algebras
    • Example: Distinguishing Two p-Groups of Class 2
• Intersections of Classical Groups
  • IsometryGroup(S : parameters): SeqEnum → GrpMat
  • Example: Isometry Group of Forms from a p-Group
  • ClassicalIntersection(S): SeqEnum → GrpMat
  • Example: Intersecting Two Classical Groups
• Bibliography