Noncommutative Associative Algebras 47
2.5.3 Universal enveloping algebras of Lie algebras
Theorem 2.5.3.1 (Poincaré–Birkhoff–Witt (PBW) theorem). If L is a fi-
nite dimensional Lie algebra over a field F with a totally ordered basis
X = {x
1
, . . . , x
n
}, then a basis of its universal associative enveloping alge-
bra U (L) consists of the monomials x
e
1
1
···x
e
n
n
with e
1
, . . . , e
n
≥ 0. Therefore:
(i) U(L) is infinite dimensional.
(ii) The canonical map α: L → U (L) is injective.
(iii) L is isomorphic to a subalgebra of the Lie algebra U(L)
−
.
Proof. By Definition 2.1.1.2, the algebra U (L) is a quotient of T (L)
∼
=
T (X).
We equip T (X) with the glex order with x
1
≺ ··· ≺ x
n
. Since both the
Lie bracket and the product in T (L)
∼
=
T (X) are bilinear, it is sufficien ...