The central augmentation ideal of a universal enveloping algebra

For a Lie algebra L over a commutative ring R, let U(L) be its universal enveloping algebra. The general Hopf-algebra construction of the intersection

C(H) = Z(H) ∩ H⁺

of the centre with the augmentation ideal, and of the ideal H C(H) it generates, lives in TauCeti.Algebra.HopfAlgebra.HopfIdeal.Augmentation. This file specialises that construction to H = U(L) by exhibiting explicit elements of it in exponential characteristic p: central p-polynomials and, for adjoint-nilpotent x, Frobenius powers of ι x.

As soon as Module.End R L is Noetherian, the central p-polynomial attached to each element of L belongs to C(U(L)); over a field, finite-dimensionality of L supplies this hypothesis. In exponential characteristic p ≠ 1, if x is nilpotent in the adjoint representation, a Frobenius power of ι x itself belongs to C(U(L)). Taking powers then puts explicit powers of ι x in every power of the generated ideal. This is the input that later makes x act nilpotently on a quotient by such a power.

Main results

• pPolynomial_ι_mem_centralAugmentation: a central p-polynomial with zero constant term belongs to the intersection.
• exists_pCentralPolynomial_mem_centralAugmentation_of_isNoetherian: under the Noetherian hypothesis, every element of L has such a p-polynomial.
• exists_pow_ι_mem_centralAugmentation_of_isNilpotent_ad: in exponential characteristic p ≠ 1, a Frobenius power of ι x lies in the intersection when ad x is nilpotent.
• exists_pow_mul_ι_mem_centralAugmentationIdeal_pow_of_isNilpotent_ad: in exponential characteristic p ≠ 1, one Frobenius exponent works in every power of the generated ideal when ad x is nilpotent.

See also

TauCeti.HopfIdeal.centralAugmentation and TauCeti.HopfIdeal.centralAugmentationIdeal, together with their two-sidedness and their finite central generating set over a left-Noetherian Hopf algebra, are in TauCeti.Algebra.HopfAlgebra.HopfIdeal.Augmentation.

References

• G. Hochschild, An Addition to Ado's Theorem, Proceedings of the American Mathematical Society 17 (1966), 531--533.

theorem TauCeti.UniversalEnvelopingAlgebra.pPolynomial_ι_mem_centralAugmentation {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {p : ℕ} (hp : p ≠ 0) (e : ℕ) (a : Fin e → R) (x : L) (hcentral : (UniversalEnvelopingAlgebra.ι R) x ^ p ^ e + ∑ i : Fin e, a i • (UniversalEnvelopingAlgebra.ι R) x ^ p ^ ↑i ∈ Subalgebra.center R (UniversalEnvelopingAlgebra R L)) : (UniversalEnvelopingAlgebra.ι R) x ^ p ^ e + ∑ i : Fin e, a i • (UniversalEnvelopingAlgebra.ι R) x ^ p ^ ↑i ∈ HopfIdeal.centralAugmentation R (UniversalEnvelopingAlgebra R L)

A central p-polynomial with zero constant term belongs to the central augmentation submodule.

theorem TauCeti.UniversalEnvelopingAlgebra.exists_pCentralPolynomial_mem_centralAugmentation_of_isNoetherian (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] (p : ℕ) [ExpChar R p] [IsNoetherian R (Module.End R L)] (x : L) : ∃ (e : ℕ) (a : Fin e → R), (UniversalEnvelopingAlgebra.ι R) x ^ p ^ e + ∑ i : Fin e, a i • (UniversalEnvelopingAlgebra.ι R) x ^ p ^ ↑i ∈ HopfIdeal.centralAugmentation R (UniversalEnvelopingAlgebra R L)

Every element has a central p-polynomial in the central augmentation submodule as soon as Module.End R L is Noetherian.

theorem TauCeti.UniversalEnvelopingAlgebra.exists_pow_ι_mem_centralAugmentation_of_isNilpotent_ad {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (p : ℕ) [ExpChar R p] (hp : p ≠ 1) {x : L} (h : IsNilpotent ((LieAlgebra.ad R L) x)) : ∃ (e : ℕ), (UniversalEnvelopingAlgebra.ι R) x ^ p ^ e ∈ HopfIdeal.centralAugmentation R (UniversalEnvelopingAlgebra R L)

If ad x is nilpotent in exponential characteristic p ≠ 1, then a Frobenius power of ι x belongs to the central augmentation submodule.

theorem TauCeti.UniversalEnvelopingAlgebra.exists_pow_mul_ι_mem_centralAugmentationIdeal_pow_of_isNilpotent_ad {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (p : ℕ) [ExpChar R p] (hp : p ≠ 1) {x : L} (h : IsNilpotent ((LieAlgebra.ad R L) x)) : ∃ (e : ℕ), ∀ (n : ℕ), (UniversalEnvelopingAlgebra.ι R) x ^ (p ^ e * n) ∈ HopfIdeal.centralAugmentationIdeal R (UniversalEnvelopingAlgebra R L) ^ n

If ad x is nilpotent in exponential characteristic p ≠ 1, every power of the central augmentation ideal contains the corresponding power of a Frobenius power of ι x.