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QUANTUM GROUPS

 
Acknowledgements
 
Introduction
 
Background
      Gaussian Binomials
      Quantized Enveloping Algebras
      Representations of Uq(L)
      PBW-type Bases
      The Z-form of Uq(L)
      The Canonical Basis
      The Path Model
 
Gauss Numbers
 
Construction
 
Related Structures
 
Operations on Elements
 
Representations
 
Hopf Algebra Structure
 
Automorphisms
 
Kashiwara Operators
 
The Path Model
 
Elements of the Canonical Basis
 
Homomorphisms to the Universal Enveloping Algebra
 
Bibliography







DETAILS

 
Introduction

 
Background

      Gaussian Binomials

      Quantized Enveloping Algebras

      Representations of Uq(L)

      PBW-type Bases

      The Z-form of Uq(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 (H102E1)

 
Construction
      QuantizedUEA(R) : RootDtm -> AlgQUE
      Example AlgQEA_QGrpConstr (H102E2)
      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 (H102E3)

 
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 (H102E4)

 
Representations
      HighestWeightRepresentation(U, w) : AlgQUE, SeqEnum -> UserProgram
      HighestWeightModule(U, w) : AlgQUE, SeqEnum -> ModTupAlg
      Example AlgQEA_QGrpEltOps (H102E5)
      WeightsAndVectors(V) : ModAlg -> SeqEnum, SeqEnum
      HighestWeightsAndVectors(V) : ModAlg -> SeqEnum, SeqEnum
      CanonicalBasis(V) : ModAlg -> SeqEnum
      Example AlgQEA_CanBasMod (H102E6)
      TensorProduct(Q) : SeqEnum -> ModAlg, Map
      Example AlgQEA_AlgQEATP (H102E7)

 
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 (H102E8)

 
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 (H102E9)

 
Kashiwara Operators
      Falpha(m, i) : AlgQUEElt, RngIntElt -> AlgQUEElt
      Ealpha(m, i) : AlgQUEElt, RngIntElt -> AlgQUEElt
      Example AlgQEA_QGrpAutoms (H102E10)

 
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 (H102E11)
      CrystalGraph(R, hw) : RootDtm, SeqEnum -> GrphDir, SeqEnum
      Example AlgQEA_CrystGrph (H102E12)

 
Elements of the Canonical Basis
      CanonicalElements(U, w) : AlgQUE, SeqEnum -> SeqEnum
      Example AlgQEA_QGrpAutoms (H102E13)
      Example AlgQEA_QGrpAutoms2 (H102E14)

 
Homomorphisms to the Universal Enveloping Algebra
      QUAToIntegralUEAMap(U) : AlgQUE -> Map
      Example AlgQEA_QEAtoUEA (H102E15)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012