pdf file

Example 2.6.6.
If A is an associative K-algebra with K a field of prime characteristic, char(K) = p, then the
commutator and the map a ↦→ a p turns it into a restricted Lie algebra.
Observation 2.6.7.
1. Let L be a restricted Lie algebra, U its universal enveloping algebra. Denote by B the
two-sided ideal in U generated by a p −a [p] for all a ∈ L. Denote by U the quotient algebra
U := U/B. It is a restricted Lie algebra with a [p] given by the p-th power.
2. Then the canonical quotient map π : L → U is a morphism of restricted Lie algebras. It
is universal in the following sense: if A is any associative algebra over K and f : L → A a
morphism of restricted Lie algebras, then there exists a unique algebra map F : U → A
such that f = F ◦ π:
L π
U
∃!F
f
A
3. By the universal property, the restricted morphisms
define algebra maps
that are uniquely determined by
L → K
a ↦→ 0
L → L × L
a ↦→ (a, a)
L → L opp
a ↦→ −a
ɛ : U → K , Δ : U → U ⊗ U and S : U → U opp
ɛ(π(a)) = 0
Δ(π(a)) = 1 ⊗ π(a) + π(a) ⊗ 1
S(π(a)) = −π(a)
for a ∈ L that turn U into a cocommutative Hopf algebra. It is called the u-algebra of
the restricted Lie algebra L.
4. One has the following analogue of the Poincaré-Birkhoff-Witt theorem: the natural map
ι L : L → U is injective. If (u i ) i∈I is an ordered basis for L, then
is a basis of U.
u k 1
i 1
∙ u k 2
i 2
. . . u kr
i r
with i 1 ≤ i 2 ≤ . . . i r and 0 ≤ k j ≤ p − 1
5. Thus if L has finite-dimension, dim K L = n, then U is finite-dimensional of dimension
dim U = p n . Thus U is a cocommutative finite-dimensional Hopf algebra. We next show
that it is not isomorphic to the group algebra of any finite group.
Definition 2.6.8
48