Poincare-Birkhoff-Witt theorem

Name
This result is called the Poincaré–Birkhoff–Witt theorem and is often abbreviated as the PBW theorem. Results similar or analogous to it are termed PBW type theorems.

Statement for algebras over fields
Suppose $$K$$ is a field and $$L$$ is a Lie algebra over $$K$$. Then, the following are true:


 * The canonical map from $$L$$ to its universal enveloping algebra $$U(L)$$ (an associative unital $$K$$-algebra) is injective.
 * The image of $$L$$ inside $$U(L)$$ generates $$U(L)$$ as an associative unital $$K$$-algebra.

Thus, every Lie algebra over a field is a Lie subalgebra of an associative unital algebra over the field, where the associative algebra itself is interpreted as a Lie algebra by the Lie bracket $$[x,y] = xy - yx$$.

Statement for algebras over commutative unital rings
Suppose $$R$$ is an (associative) commutative unital ring and $$L$$ is a Lie algebra over $$R$$. Then, the following are true:


 * The canonical map from $$L$$ to its universal enveloping algebra $$U(L)$$ (an associative unital $$R$$-algebra) is injective.
 * The image of $$L$$ inside $$U(L)$$ generates $$U(L)$$ as an associative unital $$R$$-algebra.

Thus, every Lie algebra over a commutative unital ring is a Lie subalgebra of an associative unital algebra over the same commutative unital ring, where the associative algebra itself is interpreted as a Lie algebra by the Lie bracket $$[x,y] = xy - yx$$.

Statement for Lie rings as algebras over the integers
Suppose $$L$$ is a Lie ring. Viewing $$L$$ as a Lie algebra over the ring of integers. Then, the following are true:


 * The canonical map from $$L$$ to its universal enveloping algebra $$U(L)$$ (an associative unital ring, i.e., an associative unital $$\mathbb{Z}$$-algebra) is injective.
 * The image of $$L$$ inside $$U(L)$$ generates $$U(L)$$ as an associative unital ring.