# Algebras with Involution

- [Introduction](introduction.md)

- [Algebras with Involution](star-algebras.md)

  - [Reflexive Forms](star-algebras.md#reflexive-forms)

    - [`IsometryGroup(F : parameters): AlgMatElt → GrpMat`](star-algebras.md#function-isometrygroup-algmatelt)

    - [`SimilarityGroup(F : parameters): AlgMatElt → GrpMat`](star-algebras.md#function-similaritygroup-algmatelt)

  - [Systems of Reflexive Forms](star-algebras.md#systems-of-reflexive-forms)

    - [`PGroupToForms(G): GrpPC → SeqEnum`](star-algebras.md#function-pgrouptoforms-grppc)

    - [`PGroupToForms(G,p): GrpPC, RngIntElt → SeqEnum`](star-algebras.md#function-pgrouptoforms-grppc-rngintelt)

    - [`PGroupToForms(G): GrpMat → SeqEnum`](star-algebras.md#function-pgrouptoforms-grpmat)

    - [`PGroupToForms(G,p): GrpMat, RngIntElt → SeqEnum`](star-algebras.md#function-pgrouptoforms-grpmat-rngintelt)

    - [`Example: P Group To Forms`](star-algebras.md#example-ex-32c4d8)

  - [Basic Attributes of $\ast$-Algebras](star-algebras.md#basic-attributes-of-ast-algebras)

    - [`IsStarAlgebra(A): AlgMat → BoolElt`](star-algebras.md#function-isstaralgebra-algmat)

    - [`IsStarAlgebra(A): AlgGrp → BoolElt`](star-algebras.md#function-isstaralgebra-alggrp)

    - [`Star(A): AlgMat → Map`](star-algebras.md#function-star-algmat)

    - [`Star(A): AlgGrp → Map`](star-algebras.md#function-star-alggrp)

  - [Adjoint Algebras](star-algebras.md#adjoint-algebras)

    - [`AdjointAlgebra(S : parameters): SeqEnum → AlgMat`](star-algebras.md#function-adjointalgebra-seqenum)

    - [`Example: Adjoint Algebra`](star-algebras.md#example-ex-2cfb50)

  - [Group Algebras](star-algebras.md#group-algebras)

    - [`StarOnGroupAlgebra(A): AlgGrp → Map`](star-algebras.md#function-starongroupalgebra-alggrp)

    - [`GroupAlgebraAsStarAlgebra(R, G): Rng, Grp → AlgGrp`](star-algebras.md#function-groupalgebraasstaralgebra-rng-grp)

    - [`Example: Group Algebra As Star Algebra`](star-algebras.md#example-ex-d8cda9)

    - [`Example: Group Algebra As Star Algebra2`](star-algebras.md#example-ex-fb1bbd)

  - [Simple $\ast$-Algebras](star-algebras.md#simple-ast-algebras)

    - [`SimpleStarAlgebra(name, d, K): MonStgElt, RngIntElt, FldFin → AlgMat`](star-algebras.md#function-simplestaralgebra-monstgelt-rngintelt-fldfin)

    - [`Example: Simple Star Algebra`](star-algebras.md#example-ex-bf2207)

- [Decompositions of $\ast$-Algebras](decompositions.md)

  - [`WedderburnDecomposition(A): AlgMat → AlgMat, AlgMat`](decompositions.md#function-wedderburndecomposition-algmat)

  - [`WedderburnDecomposition(A): AlgGrp → AlgGrpSub, AlgGrpSub`](decompositions.md#function-wedderburndecomposition-alggrp)

  - [`TaftDecomposition(A): AlgMat → AlgMat, AlgMat`](decompositions.md#function-taftdecomposition-algmat)

  - [`TaftDecomposition(A): AlgGrp → AlgGrpSub, AlgGrpSub`](decompositions.md#function-taftdecomposition-alggrp)

  - [`Example: Taft Decomposition`](decompositions.md#example-ex-d6a904)

- [Recognition of $\ast$-Algebras](recognition.md)

  - [Recognition of Simple $\ast$-Algebras](recognition.md#recognition-of-simple-ast-algebras)

    - [`RecogniseClassicalSSA(A): AlgMat → BoolElt, AlgMat, Map, Map`](recognition.md#function-recogniseclassicalssa-algmat)

    - [`RecogniseExchangeSSA(A): AlgMat → BoolElt, AlgMat, Map, Map`](recognition.md#function-recogniseexchangessa-algmat)

    - [`Example: Recognise Classical SSA`](recognition.md#example-ex-f5d034)

  - [Recognition of Arbitrary $\ast$-Algebras](recognition.md#recognition-of-arbitrary-ast-algebras)

    - [`RecogniseStarAlgebra(A): AlgMat → BoolElt`](recognition.md#function-recognisestaralgebra-algmat)

    - [`RecogniseStarAlgebra(A): AlgGrp → BoolElt`](recognition.md#function-recognisestaralgebra-alggrp)

    - [`IsSimpleStarAlgebra(A): AlgMat → BoolElt`](recognition.md#function-issimplestaralgebra-algmat)

    - [`IsSimpleStarAlgebra(A): AlgGrp → BoolElt`](recognition.md#function-issimplestaralgebra-alggrp)

    - [`SimpleParameters(A): AlgMat → SeqEnum`](recognition.md#function-simpleparameters-algmat)

    - [`SimpleParameters(A): AlgGrp → SeqEnum`](recognition.md#function-simpleparameters-alggrp)

    - [`NormGroup(A): AlgMat → GrpMat`](recognition.md#function-normgroup-algmat)

    - [`Example: Nonisomorphism1`](recognition.md#example-ex-9f1ea0)

    - [`Example: Nonisomorphism2`](recognition.md#example-ex-31ccc3)

- [Intersections of Classical Groups](isometry.md)

  - [`IsometryGroup(S : parameters): SeqEnum → GrpMat`](isometry.md#function-isometrygroup-seqenum)

  - [`Example: Isometry Group`](isometry.md#example-ex-575942)

  - [`ClassicalIntersection(S): SeqEnum → GrpMat`](isometry.md#function-classicalintersection-seqenum)

  - [`Example: Classical Intersection`](isometry.md#example-ex-95a8b7)