The iterated Ore presentation of U_q(sl₂) #
For a nonzero parameter q, the quantized enveloping algebra UqSl2 q is an iterated Ore
extension of the Laurent polynomial ring ℂ[K, K⁻¹]:
- first adjoin
Fwith the twistσ_F (K) = q² Kand zero derivation, giving the lower Borel partℂ[K, K⁻¹][F; σ_F], in whichF K = q² K F; - then adjoin
Ewith the twistσ_E (K) = q⁻² K,σ_E (F) = Fand theσ_E-derivationδ_E (K) = 0,δ_E (F) = (K - K⁻¹) / (q - q⁻¹), so thatE K = q⁻² K EandE F = F E + (K - K⁻¹) / (q - q⁻¹).
The twist σ_E extends from ℂ[K, K⁻¹] by the universal property of the first Ore extension.
The derivation δ_E is the upper-right corner of the algebra map
r ↦ !![σ_E r, δ_E r; 0, r] into 2 × 2 matrices, which is likewise produced by that universal
property from its values on K and F.
The two universal properties, of the presentation and of the Ore extensions, give mutually
inverse algebra maps, so UqSl2 q is isomorphic to the tower. Since an Ore extension is free on
its left monomials, this identification is the source of the PBW basis K^b F^a E^c of
UqSl2 q. Both twists are injective, so the tower, and hence UqSl2 q, is a domain; in
particular the presentation is nontrivial.
Main definitions #
TauCeti.UqSl2.sigmaF: the automorphismK ↦ q² Kofℂ[K, K⁻¹].TauCeti.UqSl2.LowerBorel: the first Ore extensionℂ[K, K⁻¹][F; σ_F].TauCeti.UqSl2.sigmaEandTauCeti.UqSl2.deltaE: the twist and skew derivation of the second extension.TauCeti.UqSl2.OreTower: the second Ore extensionℂ[K, K⁻¹][F; σ_F][E; σ_E, δ_E].TauCeti.UqSl2.oreTowerEquiv: the algebra isomorphismUqSl2 q ≃ₐ[ℂ] OreTower q.
Main results #
TauCeti.UqSl2.instIsDomain:UqSl2 qis a domain for every nonzeroq.
References #
- C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Section VI.1, for the
iterated Ore extension and the resulting PBW basis of
U_q(sl₂). - A. Virelizier, Quantum groups, Section 2.3, for the presentation conventions.
The first extension: adjoining F #
The automorphism K ↦ q² K of the Laurent polynomial ring ℂ[K, K⁻¹], the twist of the
first Ore extension of the tower presenting UqSl2 q.
Instances For
sigmaF is the rescaling K ↦ q² K.
The second extension: adjoining E #
The twist of the second Ore extension: the algebra endomorphism of LowerBorel q with
K ↦ q⁻² K and F ↦ F.
Instances For
sigmaE rescales ℂ[K, K⁻¹] by K ↦ q⁻² K.
sigmaE fixes F.
The skew derivation of the second Ore extension: the sigmaE-derivation of
LowerBorel q vanishing on ℂ[K, K⁻¹] with F ↦ (K - K⁻¹) / (q - q⁻¹).
Instances For
deltaE vanishes on ℂ[K, K⁻¹].
deltaE sends F to (K - K⁻¹) / (q - q⁻¹).
Comparison with the presentation #
oreTowerEquiv sends E to the variable of the second extension.
oreTowerEquiv sends F to the variable of the first extension.
oreTowerEquiv sends K to the Laurent variable.
oreTowerEquiv sends K⁻¹ to the inverse Laurent variable.
The inverse of oreTowerEquiv sends the variable of the second extension to E.
The inverse of oreTowerEquiv sends the variable of the first extension to F.
The Laurent polynomials in K correspond to the Laurent evaluations at the unit K.