Exercise 6, § 2

Let g be a Lie p-algebra over a ring K such that pK = {0} (p prime), U its enveloping algebra and σ the canonical mapping of g into U. Let J be the two-sided ideal of U generated by the elements (σ(x))^p − σ(x^{[p]}) where x runs through g. The associative algebra Ũ = U/J is called the restricted enveloping algebra of g. The mapping σ defines when passing to the quotient a mapping ŝ (called canonical) of g into Ũ, which is a p-homomorphism (when Ũ is considered as a Lie p-algebra).

(a) The algebra Ũ and the mapping ŝ are solutions of a universal mapping problem: for every p-homomorphism f of g into an associative algebra B over K (considered as a Lie p-algebra), there exists one and only one K-homomorphism f' of Ũ into B (with their associative algebra structures) such that f = f' ∘ ŝ.

(b) Show that, if (b_λ)_{λ ∈ Λ} is a basis of g (where Λ is totally ordered), ŝ is injective and that, if x and ŝ(x) are identified for x ∈ g, the elements Π_λ b_λ^{ν_λ} (where the λ are in increasing order, the ν_λ are zero except for a finite number and 0 ≤ ν_λ < p for all λ) form a basis of Ũ. (Canonically identifying g with a submodule of U, we write φ(x) = x^p − x^{[p]} for all x ∈ g. For every composite index α = (α_λ) ∈ N^{(L)}, let α_λ = β_λ + pγ_λ with 0 ≤ β_λ < p and let

Show that the T_α form a basis of U and the T_α such that γ = (γ_λ) ≠ 0 form a basis of J; observe that the φ(b_λ) belong to the centre of U.)