Algebras

• Introduction
  • The Categories of Algebras
• Construction of General Algebras and their Elements
  • Construction of a General Algebra
    • Algebra< R, n | Q >: Rng, RngIntElt, SeqEnum → AlgGen
    • AssociativeAlgebra< R, n | Q >: Rng, RngIntElt, SeqEnum → AlgAss
    • QuaternionAlgebra< K | a, b >: Rng, RngElt, RngElt → AlgQuat
    • LieAlgebra< R, n | Q >: Rng, RngIntElt, SeqEnum → AlgLie
    • LieAlgebra(A): AlgAss → AlgLie
    • GroupAlgebra(R, G): Rng, Grp → AlgGrp
    • MatrixAlgebra(R, n): Rng, RngInt → AlgMat
  • Construction of an Element of a General Algebra
    • Zero(A): AlgGen → AlgGenElt
    • A ! 0: AlgGen, RngIntElt → AlgGenElt
    • One(A): AlgGen → AlgGenElt
    • A ! 1: AlgGen, RngIntElt → AlgGenElt
    • Random(A): AlgGen → AlgGenElt
• Construction of Subalgebras, Ideals and Quotient Algebras
  • Subalgebras and Ideals
    • sub< A | L >: AlgGen, List → AlgGen, Map
    • lideal< A | L >: AlgGen, List → AlgGen, Map
    • rideal< A | L >: AlgGen, List → AlgGen, Map
    • ideal< A | L >: AlgGen, List → AlgGen, Map
  • Quotient Algebras
    • quo< A | L >: AlgGen, List → AlgGen, Map
    • A / S: AlgGen, AlgGen → AlgGen
• Operations on Algebras and Subalgebras
  • Invariants of an Algebra
    • CoefficientRing(A): AlgGen → Rng
    • CoefficientField(A): AlgGen → Rng
    • BaseRing(A): AlgGen → Rng
    • BaseField(A): AlgGen → Rng
    • Dimension(A): AlgGen → RngIntElt
    • # A: AlgGen → RngIntElt
  • Changing Rings
    • ChangeRing(A, S): AlgGen, Rng → AlgGen, Map
    • ChangeRing(A, S, f): AlgGen, Rng, Map → AlgGen, Map
  • Bases
    • BasisElement(A, i): AlgGen, RngIntElt → AlgGenElt
    • A . i: AlgGen, RngIntElt → AlgGenElt
    • Basis(A): AlgGen → [ AlgGenElt ]
    • IsIndependent(Q): [ AlgGen ] → BoolElt
    • ExtendBasis(S, A): AlgGen, AlgGen → [ AlgElt ]
    • ExtendBasis(Q, A): [ AlgGen ], AlgGen → [ AlgElt ]
  • Decomposition of an Algebra
    • CompositionSeries(A): AlgGen → [ AlgGen ], [ AlgGen ], AlgMatElt
    • CompositionFactors(A): AlgGen → [ AlgGen ]
    • MinimalLeftIdeals(A : parameters): AlgGen → [ AlgGen ], BoolElt
    • MinimalRightIdeals(A : parameters): AlgGen → [ AlgGen ], BoolElt
    • MinimalIdeals(A : parameters): AlgGen → [ AlgGen ], BoolElt
    • MaximalLeftIdeals(A : parameters): AlgGen → [ AlgGen ], BoolElt
    • MaximalRightIdeals(A : parameters): AlgGen → [ AlgGen ], BoolElt
    • MaximalIdeals(A : parameters): AlgGen → [ AlgGen ], BoolElt
    • JacobsonRadical(A): AlgGen → AlgGen
    • IsSemisimple(A): AlgGen → BoolElt
    • IsSimple(A): AlgGen → BoolElt
    • Example: quaternions
  • Operations on Subalgebras
    • IsZero(A): AlgGen → BoolElt
    • A eq B: AlgGen, AlgGen → BoolElt
    • A ne B: AlgGen, AlgGen → BoolElt
    • A subset B: AlgGen, AlgGen → BoolElt
    • A notsubset B: AlgGen, AlgGen → BoolElt
    • A meet B: AlgGen, AlgGen → AlgGen
    • A * B: AlgGen, AlgGen → AlgGen
    • A ^ n: AlgGen, RngIntElt → AlgGen
    • Morphism(A, B): AlgGen, AlgGen → Map
• Operations on Elements of an Algebra
  • Operations on Elements
    • a + b: AlgGenElt, AlgGenElt → AlgGenElt
    • - a: AlgGenElt → AlgGenElt
    • a - b: AlgGenElt, AlgGenElt → AlgGenElt
    • a * b: AlgGenElt, AlgGenElt → AlgGenElt
    • a * r: AlgGenElt, RngElt → AlgGenElt
    • r * a: RngElt, AlgGenElt → AlgGenElt
    • a / r: AlgGenElt, RngElt → AlgGenElt
    • a ^ n: AlgGenElt, RngIntElt → AlgGenElt
    • MinimalPolynomial(a): AlgGenElt → RngUPolElt
    • Parent(a): AlgGenElt → AlgGen
  • Comparisons and Membership
    • a eq b: AlgGenElt, AlgGenElt → BoolElt
    • a ne b: AlgGenElt, AlgGenElt → BoolElt
    • a in A: AlgGenElt, AlgGen → BoolElt
    • a notin A: AlgGenElt, AlgGen → BoolElt
  • Predicates on Elements
    • IsZero(a): AlgGenElt → BoolElt
    • IsOne(a): AlgGenElt → BoolElt
    • IsMinusOne(a): AlgGenElt → BoolElt
    • IsUnit(a): AlgGenElt → BoolElt, AlgGenElt
    • IsRegular(a): AlgGenElt → BoolElt
    • IsZeroDivisor(a): AlgGenElt → BoolElt
    • IsIdempotent(a): AlgGenElt → BoolElt
    • IsNilpotent(a): AlgGenElt → BoolElt, RngIntElt