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The lemma implies that the u-algebra of a non-trivial restricted Lie algebra cannot be
isomorphic, as a Hopf algebra, to a group algebra, since it contains no non-trivial group-like
elements.
Over fields of characteristic zero, we can recover a Lie algebra from the primitive elements
in its universal enveloping algebra:
Proposition 2.6.14.
Let g be a Lie algebra over a field K of characteristic zero with an ordered basis and ι g : g → U(g)
its universal enveloping algebra. Then
P (U(g)) = ι g (g) .
If char(K) = p, then the primitive elements of U(g) is the span of all x pk
k ≥ 0. It is a restricted Lie algebra.
with x ∈ g and
Proof.
Define
and consider the direct sum:
Since x ∈ g is primitive in U(g), we find
U n (g) := span K {x n |x ∈ g}
Δ(x n ) =
U(g) ⊃
n∑
k=0
∞⊕
U n (g) .
n=0
( n
k)
x k ⊗ x n−k .
Thus the direct sum is a subcoalgebra of U(g) and the coproduct
Δ : U(g) → U(g) ⊗ U(g)
preserves the degree. The right hand side is thereby provided with the total degree. One checks
inductively using the Poincaré-Birkhoff-Witt theorem, that the direct sum is closed under
multiplication as well (the multiplication is not homogeneous, though).
We can restrict to homogenous elements
to investigate the coproduct:
Δ(x) =
x =
l∑
j=1
l∑
λ j (x j ) n ∈ U n (g)
j=1
λ j
n∑
k=0
( n
k)
(x j ) k ⊗ (x j ) n−k
Then x is primitive, if and only if all components with bigrade (k, n − k) and 1 ≤ k ≤ n − 1
vanish. Applying multiplication, we find
( ) l∑ n
λ j (x j ) n = 0 for all k = 1, . . . , n − 1 .
k
j=1
Over a field of characteristic zero, this implies x = 0.
✷
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