Introduction

 
Background

      Gaussian Binomials

      Quantized Enveloping Algebras

      Representations of U_q(L)

      PBW-type Bases

      The 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 AlgQEA_QGrpGauss (H112E1)

 
Construction
      QuantizedUEA(R) : RootDtm -> AlgQUE
      Example AlgQEA_QGrpConstr (H112E2)
      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 AlgQEA_QGrpRelStr (H112E3)

 
Operations on Elements
      U ! 0 : AlgQUE, RngIntElt -> AlgQUEElt
      U ! 1 : AlgQUE, RngIntElt -> AlgQUEElt
      U . i : AlgQUE, RngIntElt -> AlgQUEElt
      U ! r : AlgQUE, Any -> AlgQUEElt
      KBinomial(U, i, s) : AlgQUE, RngIntElt, 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 AlgQEA_QGrpEltOps (H112E4)

 
Representations
      HighestWeightRepresentation(U, w) : AlgQUE, SeqEnum -> UserProgram
      HighestWeightModule(U, w) : AlgQUE, SeqEnum -> ModTupAlg
      Example AlgQEA_QGrpEltOps (H112E5)
      WeightsAndVectors(V) : ModAlg -> SeqEnum, SeqEnum
      HighestWeightsAndVectors(V) : ModAlg -> SeqEnum, SeqEnum
      CanonicalBasis(V) : ModAlg -> SeqEnum
      Example AlgQEA_CanBasMod (H112E6)
      TensorProduct(Q) : SeqEnum -> ModAlg, Map
      Example AlgQEA_AlgQEATP (H112E7)

 
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 AlgQEA_QGrpComult (H112E8)

 
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
      Example AlgQEA_QGrpAutoms (H112E9)

 
Kashiwara Operators
      Falpha(m, i) : AlgQUEElt, RngIntElt -> AlgQUEElt
      Ealpha(m, i) : AlgQUEElt, RngIntElt -> AlgQUEElt
      Example AlgQEA_QGrpAutoms (H112E10)

 
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 AlgQEA_LSPaths (H112E11)
      CrystalGraph(R, hw) : RootDtm, SeqEnum -> GrphDir, SeqEnum
      Example AlgQEA_CrystGrph (H112E12)

 
Elements of the Canonical Basis
      CanonicalElements(U, w) : AlgQUE, SeqEnum -> SeqEnum
      Example AlgQEA_QGrpAutoms (H112E13)
      Example AlgQEA_QGrpAutoms2 (H112E14)

 
Homomorphisms to the Universal Enveloping Algebra
      QUAToIntegralUEAMap(U) : AlgQUE -> Map
      Example AlgQEA_QEAtoUEA (H112E15)

 
Bibliography

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