Crossed module over a Lie ring

Definition
Suppose $$M$$ and $$N$$ are Lie rings. We say that $$N$$ is a crossed module over $$M$$ if there exists a homomorphism $$\alpha:M \to \operatorname{Der}(N)$$ and a homomorphism $$\mu:N \to M$$ such that the following hold:


 * $$\mu(\alpha(m)(n)) = [m,\mu(n)] \ \forall \ m \in M, n \in N$$
 * $$\alpha(\mu(n_1))(n_2) = [n_1,n_2] \ \forall \ n_1,n_2 \in N$$

If we use $$\cdot$$ to denote the action by derivations and also the left adjoint action, then the above read as:


 * $$\mu(m \cdot n) = m \cdot \mu(n) \ \forall \ m \in M, n \in N$$
 * $$\mu(n_1) \cdot n_2 = n_1 \cdot n_2 \ \forall \ n_1,n_2 \in N$$

Facts

 * Crossed module over a Lie ring defines a compatible pair of actions of Lie rings

Related notions

 * Crossed module: The corresponding notion for groups.