The center of U_q(sl₂)

At a generic parameter q (nonzero and not a root of unity), the center of UqSl2 q is the polynomial algebra ℂ[Ω] generated by the Casimir element Ω.

The proof is a leading-coefficient reduction in the PBW basis F ^ a * K ^ b * E ^ c, using the coordinate reductions from TauCeti.QuantumGroups.UqSl2.PBW.

• Conjugation by K multiplies F ^ a * K ^ b * E ^ c by q ^ (2 (c - a)), so an element commuting with K only involves monomials with a = c.
• Commuting E past F ^ a * K ^ b * E ^ c produces q ^ (-2 b) F ^ a K ^ b E ^ (c + 1) plus monomials whose E-exponent is still c. Comparing coefficients of the monomials with the largest E-exponent n, an element commuting with E has no monomial F ^ a * K ^ b * E ^ n with b ≠ 0.
• The same computation shows that Ω ^ n has E-exponents at most n and coefficient 1 on F ^ n * E ^ n. Subtracting a multiple of Ω ^ n from a central element lowers its largest E-exponent, and induction finishes the proof.

The same leading coefficients show that the powers of Ω are linearly independent, so Ω is transcendental over ℂ for every nonzero q.

Main results

• TauCeti.UqSl2.transcendental_casimir: the Casimir element is transcendental over ℂ.
• TauCeti.UqSl2.center_eq_adjoin_casimir: at a generic parameter, the center of UqSl2 q is the subalgebra generated by the Casimir element.
• TauCeti.UqSl2.polynomialAlgEquivCenter: the resulting isomorphism ℂ[X] ≃ₐ[ℂ] Z(U_q(sl₂)) sending X to Ω.

References

• C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Section VI.4, for the description of the center at a generic parameter; Kassel argues through the Harish-Chandra homomorphism instead of the leading-coefficient reduction used here.

The Casimir element on PBW monomials

Powers of the Casimir element

theorem TauCeti.UqSl2.transcendental_casimir {q : ℂ} [Fact (q ≠ 0)] : Transcendental ℂ (casimir q)

The Casimir element is transcendental over ℂ: its powers are linearly independent.

The center

theorem TauCeti.UqSl2.center_eq_adjoin_casimir {q : ℂ} (hq : IsGenericQuantumParameter q) : Subalgebra.center ℂ (UqSl2 q) = ℂ[casimir q]

The center of U_q(sl₂). At a generic parameter, the center of UqSl2 q is the subalgebra generated by the Casimir element.

noncomputable def TauCeti.UqSl2.polynomialAlgEquivCenter {q : ℂ} (hq : IsGenericQuantumParameter q) : Polynomial ℂ ≃ₐ[ℂ] ↥(Subalgebra.center ℂ (UqSl2 q))

At a generic parameter, the center of U_q(sl₂) is a polynomial algebra in one variable: evaluation at the Casimir element is an isomorphism ℂ[X] ≃ₐ[ℂ] Z(U_q(sl₂)).

@[simp]

theorem TauCeti.UqSl2.coe_polynomialAlgEquivCenter_apply {q : ℂ} (hq : IsGenericQuantumParameter q) (p : Polynomial ℂ) : ↑((polynomialAlgEquivCenter hq) p) = (Polynomial.aeval (casimir q)) p

The isomorphism polynomialAlgEquivCenter is evaluation at the Casimir element.