Module over a Lie ring

Definition

Symbol-free definition

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

Let $L$ be a Lie ring and $A$ be an abelian group. A module structure of $A$ over $L$ is a map:

$\cdot :L\times A\to A$

satisfying the following:

1. For every $l\in L$, the map $a\mapsto l\cdot a$ is an endomorphism of the abelian group $A$.
2. For every $a\in A$, the map $l\mapsto l\cdot a$ is a homomorphism of groups from the additive group of $L$ to $A$.
3. For $l_{1},l_{2}\in L$ and $a\in A$, we have $[l_{1},l_{2}]\cdot a=l_{1}\cdot (l_{2}\cdot a)-l_{2}\cdot (l_{1}\cdot a)$.

Facts

• 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.