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.