Quantum Groups

• Introduction
• Background
  • Gaussian Binomials
  • Quantized Enveloping Algebras
  • Representations of \(U_q(L)\)
  • PBW-type Bases
  • The \({\mathbb{Z}}\)-form of \(U_q(L)\)
  • The Canonical Basis
  • The Path Model
• Gauss Numbers
  • GaussNumber(n, v): RngIntElt, RngElt → RngElt
  • GaussianFactorial(n, v): RngIntElt, RngElt → RngElt
  • GaussianBinomial(n, k, v): RngIntElt, RngIntElt, RngElt → RngElt
  • Example: Q Grp Gauss
• Construction
  • QuantizedUEA(R): RootDtm → AlgQUE
  • QuantizedUEAlgebra(R): RootDtm → AlgQUE
  • QuantizedUniversalEnvelopingAlgebra(R): RootDtm → AlgQUE
  • Example: Q Grp Constr
  • AssignNames(U, S): AlgPBW, [ MonStgElt ]
  • ChangeRing(U, R): AlgQUE, Rng → AlgQUE
• Related Structures
  • CoefficientRing(U): AlgQUE → Fld
  • RootDatum(U): AlgQUE → RootDtm
  • PositiveRootsPerm(U): AlgQUE → SeqEnum
  • Example: Q Grp Rel Str
• Operations on Elements
  • x + y: AlgQUEElt, AlgQUEElt → AlgQUEElt
  • x - y: AlgQUEElt, AlgQUEElt → AlgQUEElt
  • x * y: AlgQUEElt, AlgQUEElt → AlgQUEElt
  • c * x: RngElt, AlgQUEElt → AlgQUEElt
  • x * c: AlgQUEElt, RngElt → AlgQUEElt
  • x ^ n: AlgQUEElt, RngIntElt → AlgQUEElt
  • U ! 0: AlgQUE, RngIntElt → AlgQUEElt
  • Zero(U): AlgQUE → AlgQUEElt
  • U ! 1: AlgQUE, RngIntElt → AlgQUEElt
  • One(U): AlgQUE → AlgQUEElt
  • U . i: AlgQUE, RngIntElt → AlgQUEElt
  • U ! r: AlgQUE, Any → AlgQUEElt
  • KBinomial(U, i, s): AlgQUE, RngIntElt, RngIntElt → AlgQUEElt
  • KBinomial(K, s): AlgQUEElt, RngIntElt → AlgQUEElt
  • Monomials(u): AlgQUEElt → SeqEnum
  • Coefficients(u): AlgQUEElt → SeqEnum
  • K ^ -1: AlgQUEElt, RngIntElt → AlgQUEElt
  • Degree(u, i): AlgQUEElt, RngIntElt → RngIntElt
  • KDegree(m, i): AlgQUEElt, RngIntElt → Tup
  • Example: Q Grp Elt Ops
• Representations
  • HighestWeightRepresentation(U, w): AlgQUE, SeqEnum → UserProgram
  • HighestWeightModule(U, w): AlgQUE, SeqEnum → ModTupAlg
  • Example: Q Grp Elt Ops
  • WeightsAndVectors(V): ModAlg → SeqEnum, SeqEnum
  • HighestWeightsAndVectors(V): ModAlg → SeqEnum, SeqEnum
  • CanonicalBasis(V): ModAlg → SeqEnum
  • Example: Can Bas Mod
  • TensorProduct(Q): SeqEnum → ModAlg, Map
  • Example: Alg QEATP
• Hopf Algebra Structure
  • UseTwistedHopfStructure(U, f, g): AlgQUE, Map, Map
  • HasTwistedHopfStructure(U): AlgQUE → BoolElt, List
  • Counit(U): AlgQUE → Map
  • Antipode(U): AlgQUE → Map
  • Comultiplication(U, d): AlgQUE, RngIntElt → UserProgram
  • Example: Q Grp Comult
• Automorphisms
  • BarAutomorphism(U): AlgQUE → Map
  • AutomorphismOmega(U): AlgQUE → Map
  • AntiAutomorphismTau(U): AlgQUE → Map
  • AutomorphismTalpha(U, k): AlgQUE, RngIntElt → Map
  • DiagramAutomorphism(U, p): AlgQUE, GrpPermElt → Map
  • GraphAutomorphism(U, p): AlgQUE, GrpPermElt → Map
  • Example: Q Grp Automs
• Kashiwara Operators
  • Falpha(m, i): AlgQUEElt, RngIntElt → AlgQUEElt
  • Ealpha(m, i): AlgQUEElt, RngIntElt → AlgQUEElt
  • Example: Q Grp Automs
• The Path Model
  • DominantLSPath(R, hw): RootDtm, SeqEnum → PathLS
  • Falpha(p, i): PathLS, RngIntElt → PathLS
  • Ealpha(p, i): PathLS, RngIntElt → PathLS
  • WeightSequence(p): PathLS → SeqEnum
  • RationalSequence(p): PathLS → SeqEnum
  • EndpointWeight(p): PathLS → ModTupRngElt
  • Shape(p): PathLS → ModTupRngElt
  • WeylWord(p): PathLS → SeqEnum
  • IsZero(p): PathLS → BoolElt
  • p1 eq p2: PathLS, PathLS → BoolElt
  • Example: LS Paths
  • CrystalGraph(R, hw): RootDtm, SeqEnum → GrphDir, SeqEnum
  • Example: Cryst Grph
• Elements of the Canonical Basis
  • CanonicalElements(U, w): AlgQUE, SeqEnum → SeqEnum
  • Example: Q Grp Automs
  • Example: Q Grp Automs2
• Homomorphisms to the Universal Enveloping Algebra
  • QUAToIntegralUEAMap(U): AlgQUE → Map
  • Example: QE Ato UEA