Free Lie ring on an abelian group
Let be an abelian group. The free Lie ring on is defined as a Lie ring along with an embedding of as a subgroup of the additive group of such that, for any Lie ring and any group homomorphism from to extends uniquely to a homomorphism of Lie rings from to .
Note that this notion is quite different from that of the associated Lie ring of a group. The associated Lie ring, when applied to an abelian group , would give an abelian Lie ring whose additive group is precisely . The free Lie ring, on the other hand, would have separate elements for various brackets of things in , set to be equal only based on the rules of addition in and the identities that Lie rings must satisfy.