# Quantum Groups

- [Introduction](introduction.md)

- [Background](background.md)

  - [Gaussian Binomials](background.md#gaussian-binomials)

  - [Quantized Enveloping Algebras](background.md#quantized-enveloping-algebras)

  - [Representations of $U_q(L)$](background.md#representations-of-u-q-l)

  - [PBW-type Bases](background.md#pbw-type-bases)

  - [The ${\mathbb{Z}}$-form of $U_q(L)$](background.md#the-mathbb-z-form-of-u-q-l)

  - [The Canonical Basis](background.md#the-canonical-basis)

  - [The Path Model](background.md#the-path-model)

- [Gauss Numbers](gauss.md)

  - [`GaussNumber(n, v): RngIntElt, RngElt → RngElt`](gauss.md#function-gaussnumber-rngintelt-rngelt)

  - [`GaussianFactorial(n, v): RngIntElt, RngElt → RngElt`](gauss.md#function-gaussianfactorial-rngintelt-rngelt)

  - [`GaussianBinomial(n, k, v): RngIntElt, RngIntElt, RngElt → RngElt`](gauss.md#function-gaussianbinomial-rngintelt-rngintelt-rngelt)

  - [`Example: Q Grp Gauss`](gauss.md#example-ex-331222)

- [Construction](construct.md)

  - [`QuantizedUEA(R): RootDtm → AlgQUE`](construct.md#function-quantizeduea-rootdtm)

  - [`QuantizedUEAlgebra(R): RootDtm → AlgQUE`](construct.md#function-quantizeduealgebra-rootdtm)

  - [`QuantizedUniversalEnvelopingAlgebra(R): RootDtm → AlgQUE`](construct.md#function-quantizeduniversalenvelopingalgebra-rootdtm)

  - [`Example: Q Grp Constr`](construct.md#example-ex-578e8e)

  - [`AssignNames(U, S): AlgPBW, [ MonStgElt ]`](construct.md#function-assignnames-algpbw-monstgelt)

  - [`ChangeRing(U, R): AlgQUE, Rng → AlgQUE`](construct.md#function-changering-algque-rng)

- [Related Structures](rel-struct.md)

  - [`CoefficientRing(U): AlgQUE → Fld`](rel-struct.md#function-coefficientring-algque)

  - [`RootDatum(U): AlgQUE → RootDtm`](rel-struct.md#function-rootdatum-algque)

  - [`PositiveRootsPerm(U): AlgQUE → SeqEnum`](rel-struct.md#function-positiverootsperm-algque)

  - [`Example: Q Grp Rel Str`](rel-struct.md#example-ex-3ae424)

- [Operations on Elements](elts-ops.md)

  - [`x + y: AlgQUEElt, AlgQUEElt → AlgQUEElt`](elts-ops.md#operation-op-plus-algqueelt-algqueelt)

  - [`x - y: AlgQUEElt, AlgQUEElt → AlgQUEElt`](elts-ops.md#operation-op-minus-algqueelt-algqueelt)

  - [`x * y: AlgQUEElt, AlgQUEElt → AlgQUEElt`](elts-ops.md#operation-op-times-algqueelt-algqueelt)

  - [`c * x: RngElt, AlgQUEElt → AlgQUEElt`](elts-ops.md#operation-op-times-rngelt-algqueelt)

  - [`x * c: AlgQUEElt, RngElt → AlgQUEElt`](elts-ops.md#operation-op-times-algqueelt-rngelt)

  - [`x ^ n: AlgQUEElt, RngIntElt → AlgQUEElt`](elts-ops.md#operation-op-pow-algqueelt-rngintelt)

  - [`U ! 0: AlgQUE, RngIntElt → AlgQUEElt`](elts-ops.md#operation-op-algque-rngintelt)

  - [`Zero(U): AlgQUE → AlgQUEElt`](elts-ops.md#function-zero-algque)

  - [`U ! 1: AlgQUE, RngIntElt → AlgQUEElt`](elts-ops.md#operation-op-algque-rngintelt-2)

  - [`One(U): AlgQUE → AlgQUEElt`](elts-ops.md#function-one-algque)

  - [`U . i: AlgQUE, RngIntElt → AlgQUEElt`](elts-ops.md#operation-operation-algque-rngintelt-algqueelt)

  - [`U ! r: AlgQUE, Any → AlgQUEElt`](elts-ops.md#operation-op-algque-any)

  - [`KBinomial(U, i, s): AlgQUE, RngIntElt, RngIntElt → AlgQUEElt`](elts-ops.md#function-kbinomial-algque-rngintelt-rngintelt)

  - [`KBinomial(K, s): AlgQUEElt, RngIntElt → AlgQUEElt`](elts-ops.md#function-kbinomial-algqueelt-rngintelt)

  - [`Monomials(u): AlgQUEElt → SeqEnum`](elts-ops.md#function-monomials-algqueelt)

  - [`Coefficients(u): AlgQUEElt → SeqEnum`](elts-ops.md#function-coefficients-algqueelt)

  - [`K ^ -1: AlgQUEElt, RngIntElt → AlgQUEElt`](elts-ops.md#operation-op-pow-algqueelt-rngintelt-2)

  - [`Degree(u, i): AlgQUEElt, RngIntElt → RngIntElt`](elts-ops.md#function-degree-algqueelt-rngintelt)

  - [`KDegree(m, i): AlgQUEElt, RngIntElt → Tup`](elts-ops.md#function-kdegree-algqueelt-rngintelt)

  - [`Example: Q Grp Elt Ops`](elts-ops.md#example-ex-a73ba4)

- [Representations](representations.md)

  - [`HighestWeightRepresentation(U, w): AlgQUE, SeqEnum → UserProgram`](representations.md#function-highestweightrepresentation-algque-seqenum)

  - [`HighestWeightModule(U, w): AlgQUE, SeqEnum → ModTupAlg`](representations.md#function-highestweightmodule-algque-seqenum)

  - [`Example: Q Grp Elt Ops`](representations.md#example-ex-723df4)

  - [`WeightsAndVectors(V): ModAlg → SeqEnum, SeqEnum`](representations.md#function-weightsandvectors-modalg)

  - [`HighestWeightsAndVectors(V): ModAlg → SeqEnum, SeqEnum`](representations.md#function-highestweightsandvectors-modalg)

  - [`CanonicalBasis(V): ModAlg → SeqEnum`](representations.md#function-canonicalbasis-modalg)

  - [`Example: Can Bas Mod`](representations.md#example-ex-36468d)

  - [`TensorProduct(Q): SeqEnum → ModAlg, Map`](representations.md#function-tensorproduct-seqenum)

  - [`Example: Alg QEATP`](representations.md#example-ex-e51b84)

- [Hopf Algebra Structure](hopf-structure.md)

  - [`UseTwistedHopfStructure(U, f, g): AlgQUE, Map, Map`](hopf-structure.md#function-usetwistedhopfstructure-algque-map-map)

  - [`HasTwistedHopfStructure(U): AlgQUE → BoolElt, List`](hopf-structure.md#function-hastwistedhopfstructure-algque)

  - [`Counit(U): AlgQUE → Map`](hopf-structure.md#function-counit-algque)

  - [`Antipode(U): AlgQUE → Map`](hopf-structure.md#function-antipode-algque)

  - [`Comultiplication(U, d): AlgQUE, RngIntElt → UserProgram`](hopf-structure.md#function-comultiplication-algque-rngintelt)

  - [`Example: Q Grp Comult`](hopf-structure.md#example-ex-a3845c)

- [Automorphisms](automorphisms.md)

  - [`BarAutomorphism(U): AlgQUE → Map`](automorphisms.md#function-barautomorphism-algque)

  - [`AutomorphismOmega(U): AlgQUE → Map`](automorphisms.md#function-automorphismomega-algque)

  - [`AntiAutomorphismTau(U): AlgQUE → Map`](automorphisms.md#function-antiautomorphismtau-algque)

  - [`AutomorphismTalpha(U, k): AlgQUE, RngIntElt → Map`](automorphisms.md#function-automorphismtalpha-algque-rngintelt)

  - [`DiagramAutomorphism(U, p): AlgQUE, GrpPermElt → Map`](automorphisms.md#function-diagramautomorphism-algque-grppermelt)

  - [`GraphAutomorphism(U, p): AlgQUE, GrpPermElt → Map`](automorphisms.md#function-graphautomorphism-algque-grppermelt)

  - [`Example: Q Grp Automs`](automorphisms.md#example-ex-f34802)

- [Kashiwara Operators](kashiwara-ops.md)

  - [`Falpha(m, i): AlgQUEElt, RngIntElt → AlgQUEElt`](kashiwara-ops.md#function-falpha-algqueelt-rngintelt)

  - [`Ealpha(m, i): AlgQUEElt, RngIntElt → AlgQUEElt`](kashiwara-ops.md#function-ealpha-algqueelt-rngintelt)

  - [`Example: Q Grp Automs`](kashiwara-ops.md#example-ex-eb13a8)

- [The Path Model](path-model.md)

  - [`DominantLSPath(R, hw): RootDtm, SeqEnum → PathLS`](path-model.md#function-dominantlspath-rootdtm-seqenum)

  - [`Falpha(p, i): PathLS, RngIntElt → PathLS`](path-model.md#function-falpha-pathls-rngintelt)

  - [`Ealpha(p, i): PathLS, RngIntElt → PathLS`](path-model.md#function-ealpha-pathls-rngintelt)

  - [`WeightSequence(p): PathLS → SeqEnum`](path-model.md#function-weightsequence-pathls)

  - [`RationalSequence(p): PathLS → SeqEnum`](path-model.md#function-rationalsequence-pathls)

  - [`EndpointWeight(p): PathLS → ModTupRngElt`](path-model.md#function-endpointweight-pathls)

  - [`Shape(p): PathLS → ModTupRngElt`](path-model.md#function-shape-pathls)

  - [`WeylWord(p): PathLS → SeqEnum`](path-model.md#function-weylword-pathls)

  - [`IsZero(p): PathLS → BoolElt`](path-model.md#function-iszero-pathls)

  - [`p1 eq p2: PathLS, PathLS → BoolElt`](path-model.md#operation-op-eq-pathls-pathls)

  - [`Example: LS Paths`](path-model.md#example-ex-cccc8c)

  - [`CrystalGraph(R, hw): RootDtm, SeqEnum → GrphDir, SeqEnum`](path-model.md#function-crystalgraph-rootdtm-seqenum)

  - [`Example: Cryst Grph`](path-model.md#example-ex-3f9751)

- [Elements of the Canonical Basis](canbas.md)

  - [`CanonicalElements(U, w): AlgQUE, SeqEnum → SeqEnum`](canbas.md#function-canonicalelements-algque-seqenum)

  - [`Example: Q Grp Automs`](canbas.md#example-ex-60f4ee)

  - [`Example: Q Grp Automs2`](canbas.md#example-ex-7c2231)

- [Homomorphisms to the Universal Enveloping Algebra](q.md)

  - [`QUAToIntegralUEAMap(U): AlgQUE → Map`](q.md#function-quatointegralueamap-algque)

  - [`Example: QE Ato UEA`](q.md#example-ex-08caa9)