Separating the canonical copy of a Lie algebra by a finite-dimensional quotient

Let L be a finite-dimensional Lie algebra over a field K of prime characteristic p and write UC for the two-sided ideal of U(L) generated by the central elements of augmentation zero. Each element of a basis of L has a central p-polynomial, and those finitely many central elements generate a commutative subalgebra over which U(L) is a finite module, so the centre of U(L) is Noetherian and U(L) is finite over it. Those are the hypotheses under which the generalized Krull intersection theorem applies to UC, and when U(L) has no zero divisors it gives

⋂ₙ UC ^ n = 0,

because an element of every power is fixed by a central multiplier of augmentation zero, whose complement to 1 has augmentation 1 and is therefore nonzero.

The canonical copy of L in U(L) is finite dimensional, so its traces on the filtration UC ^ n cannot descend forever: a single power already meets it only in the image of 0. Since U(L) ⧸ UC ^ n is finite dimensional, left multiplication on that quotient is a finite-dimensional representation of L whose kernel is exactly the kernel of ι : L → U(L), and in which every element whose adjoint action on L is nilpotent acts nilpotently — Hochschild's strengthening, which for this construction holds at every stage of the filtration.

Injectivity of ι therefore turns the representation below into a faithful one, which is the Ado--Iwasawa theorem in prime characteristic; over a field that injectivity is a corollary of the Poincaré--Birkhoff--Witt theorem, which also supplies the absence of zero divisors in U(L) through the symmetric associated graded algebra. Both properties hold for every Lie algebra over a field, by TauCeti.UniversalEnvelopingAlgebra.instIsDomain and TauCeti.UniversalEnvelopingAlgebra.ι_injective in the PBW development.

Main results

• TauCeti.UniversalEnvelopingAlgebra.iInf_centralAugmentationIdeal_pow_eq_bot: the powers of the central augmentation ideal of U(L) meet in zero.
• TauCeti.UniversalEnvelopingAlgebra.exists_forall_ι_eq_zero_of_mem_centralAugmentationIdeal_pow: some power of the central augmentation ideal meets the canonical copy of L only in zero.
• TauCeti.UniversalEnvelopingAlgebra.exists_algHom_eq_zero_iff_ι_eq_zero: a finite-dimensional algebra quotient of U(L) whose kernel meets the canonical copy of L in the kernel of ι, and in which every adjoint-nilpotent element of L becomes nilpotent.
• TauCeti.UniversalEnvelopingAlgebra.exists_representation_eq_zero_iff_ι_eq_zero: the corresponding finite-dimensional representation of L, by left multiplication on that quotient.

References

• G. Hochschild, An Addition to Ado's Theorem, Proceedings of the American Mathematical Society 17 (1966), 531--533.
• N. Jacobson, Lie Algebras, Interscience (1962), Chapter VI and pp. 202--203.

theorem TauCeti.UniversalEnvelopingAlgebra.iInf_centralAugmentationIdeal_pow_eq_bot (K : Type u) (L : Type v) [Field K] [LieRing L] [LieAlgebra K L] [FiniteDimensional K L] (p : ℕ) [Fact (Nat.Prime p)] [CharP K p] : ⨅ (n : ℕ), HopfIdeal.centralAugmentationIdeal K (UniversalEnvelopingAlgebra K L) ^ n = ⊥

The powers of the central augmentation ideal of an enveloping algebra meet in zero. In prime characteristic the centre of U(L) is Noetherian and U(L) is a finite module over it, so the generalized Krull intersection theorem applies: an element of every power is fixed by a central multiplier of augmentation zero, and the complement to 1 of such a multiplier has augmentation 1, hence is nonzero.

theorem TauCeti.UniversalEnvelopingAlgebra.exists_forall_ι_eq_zero_of_mem_centralAugmentationIdeal_pow (K : Type u) (L : Type v) [Field K] [LieRing L] [LieAlgebra K L] [FiniteDimensional K L] (p : ℕ) [Fact (Nat.Prime p)] [CharP K p] : ∃ (n : ℕ), ∀ (x : L), (UniversalEnvelopingAlgebra.ι K) x ∈ HopfIdeal.centralAugmentationIdeal K (UniversalEnvelopingAlgebra K L) ^ n → (UniversalEnvelopingAlgebra.ι K) x = 0

A single power of the central augmentation ideal already separates the canonical copy of L. The canonical copy is a finite-dimensional subspace of U(L), so its traces on the descending filtration by the powers UC ^ n stabilize at its trace on their intersection, which is zero.

theorem TauCeti.UniversalEnvelopingAlgebra.exists_algHom_eq_zero_iff_ι_eq_zero (K : Type u) (L : Type v) [Field K] [LieRing L] [LieAlgebra K L] [FiniteDimensional K L] (p : ℕ) [Fact (Nat.Prime p)] [CharP K p] : ∃ (A : Type (max u v)) (x : Ring A) (x_1 : Algebra K A) (_ : FiniteDimensional K A) (q : UniversalEnvelopingAlgebra K L →ₐ[K] A), (∀ (x_3 : L), q ((UniversalEnvelopingAlgebra.ι K) x_3) = 0 ↔ (UniversalEnvelopingAlgebra.ι K) x_3 = 0) ∧ ∀ (x_3 : L), IsNilpotent ((LieAlgebra.ad K L) x_3) → IsNilpotent (q ((UniversalEnvelopingAlgebra.ι K) x_3))

The canonical copy of L in U(L) is separated by a finite-dimensional algebra quotient, which moreover makes every adjoint-nilpotent element nilpotent. The quotient is U(L) ⧸ UC ^ n for a power UC ^ n that meets the canonical copy of L only in zero.

With the injectivity of ι, which over a field is a corollary of the Poincaré--Birkhoff--Witt theorem, this is a finite-dimensional associative target of U(L) separating the canonical copy of L, the input from which a faithful finite-dimensional representation of L is built.

theorem TauCeti.UniversalEnvelopingAlgebra.exists_representation_eq_zero_iff_ι_eq_zero (K : Type u) (L : Type v) [Field K] [LieRing L] [LieAlgebra K L] [FiniteDimensional K L] (p : ℕ) [Fact (Nat.Prime p)] [CharP K p] : ∃ (V : Type (max u v)) (x : AddCommGroup V) (x_1 : Module K V) (_ : FiniteDimensional K V) (ρ : L →ₗ⁅K⁆ Module.End K V), (∀ (x_3 : L), ρ x_3 = 0 ↔ (UniversalEnvelopingAlgebra.ι K) x_3 = 0) ∧ ∀ (x_3 : L), IsNilpotent ((LieAlgebra.ad K L) x_3) → IsNilpotent (ρ x_3)

A finite-dimensional representation of L whose kernel is the kernel of ι : L → U(L) and which carries adjoint-nilpotent elements to nilpotent endomorphisms. It is the left-regular action of L on the separating finite-dimensional quotient of U(L).

Injectivity of ι, which over a field is a corollary of the Poincaré--Birkhoff--Witt theorem, makes this representation faithful; it is then Hochschild's strengthening of the Ado--Iwasawa theorem in prime characteristic.