Structure Constant Algebras

• Introduction
• Construction of Structure Constant Algebras and Elements
  • Construction of a Structure Constant Algebra
    • Algebra< R, n | Q : parameters >: Rng, RngIntElt, SeqEnum → AlgGen
    • Algebra< M | Q : parameters >: ModTupRng, SeqEnum → AlgGen
    • Algebra< R, n | T : parameters >: Rng, RngIntElt, SeqEnum → AlgGen
    • ChangeBasis(A, B): AlgGen, {[AlgGenElt]} → AlgGen
    • ChangeBasis(A, B): AlgGen, {[ModTupFldElt]} → AlgGen
    • ChangeBasis(A, B): AlgGen, Mtrx → AlgGen
  • Construction of Elements of a Structure Constant Algebra
    • elt< A | r₁, r₂, ..., rₙ >: AlgGen, RngElt, RngElt, ..., RngElt → AlgGenElt
    • A ! Q: AlgGen, SeqEnum[RngElt] → AlgGenElt
    • BasisProduct(A, i, j): AlgGen, RngIntElt, RngIntElt → AlgGenElt
    • BasisProducts(A): AlgGen → SeqEnum
• Operations on Structure Constant Algebras and Elements
  • Operations on Structure Constant Algebras
    • IsCommutative(A): AlgGen → BoolElt
    • IsAssociative(A): AlgGen → BoolElt
    • IsLie(A): AlgGen → BoolElt
    • DirectSum(A, B): AlgGen, AlgGen → AlgGen
    • Example: Jordan Algebra from Structure Constants
  • Indexing Elements
    • a[i]: AlgGenElt, RngIntElt → RngElt
    • a[i] := r: AlgGenElt, RngIntElt, RngElt → AlgGenElt
  • The Module Structure of a Structure Constant Algebra
    • Module(A): AlgGen → ModTupRng
    • Degree(A): AlgGen → RngIntElt
    • Degree(a): AlgGenElt → RngIntElt
    • ElementToSequence(a): AlgGenElt → SeqEnum
    • Eltseq(a): AlgGenElt → SeqEnum
    • Coordinates(S, a): AlgGen, AlgGenElt → SeqEnum
    • InnerProduct(a, b): AlgGenElt, AlgGenElt → RngElt
    • Support(a): AlgGenElt → SetEnum
  • Homomorphisms
    • hom< A -> B | Q >: AlgGen, AlgGen, [ AlgGenElt ] → Map
    • hom< A -> B | Q >: AlgGen, TupMod, [ TupModElt ] → Map
    • Example: Real Cayley Algebra