Module over a Lie ring
A module over a Lie ring is an abelian group along with a homomorphism of Lie rings from the Lie ring to the endomorphism ring of the abelian group, with the Lie bracket given by the additive commutator.
Definition with symbols
satisfying the following:
- For every , the map is an endomorphism of the abelian group .
- For every , the map is a homomorphism of groups from the additive group of to .
- For and , we have .
- The notion of module over a Lie ring is closely related to the notion of group action. While groups act on sets, Lie rings act on abelian groups.
- Just as we can have a group acting on another group by automorphisms, we can have a module over a Lie ring that is itself a Lie ring, where every element of the acting Lie ring acts as derivations. This is equivalent to a homomorphism from the acting Lie ring to the Lie ring of derivations of the Lie ring being acted upon.
- Just as a group acts as automorphisms by conjugation, i.e., via inner automorphisms, we have that a Lie ring acts as derivations by adjoint action.