1-cocycle for a Lie ring action

Definition

Suppose $L$ is a Lie ring, $M$ is an abelian group, and $\varphi:L \to \operatorname{End}(M)$ is a Lie ring homomorphism from $L$ to the ring of endomorphisms of $M$ , where the latter gets the usual Lie ring structure from its structure as an associative ring.

Explicit definition

A 1-cocycle for this action is a homomorphism of groups $f:L \to M$ satisfying the additional condition:

$\! f([x,y]) = \varphi(f(x))(y) - \varphi(f(y))(x) \ \forall \ x,y \in L$

Equivalently:

$\! \varphi(f(x))(y) - \varphi(f(y))(x) - f([x,y]) = 0 \ \forall \ x,y \in L$

If we suppress $\varphi$ and denote the action by $\cdot$ , this can be written as:

$\! f([x,y]) = f(x) \cdot y - f(y) \cdot x \ \forall \ x,y \in L$

Definition in terms of the general definition of cocycle

A 1-cocycle for a Lie ring action is a special case of a cocycle for a Lie ring action, namely $n = 1$ .

Group structure

The set of 1-cocycles for the action of $L$ on $M$ forms a group under pointwise addition.

Particular cases and variations

Case or variation	Condition for an additive homomorphism to be a 1-cocycle	Further information
$M = L$ and $\varphi$ is the adjoint action, i.e., $\varphi(x)$ is the inner derivation induced by $x$	$\! f([x,y]) = [f(x),y] + [x,f(y)]$	Also called derivation of a Lie ring
$\varphi$ is the zero map	$f$ is the zero map on all of $[L,L]$ , i.e., it descends to a homomorphism from $L/[L,L]$
$L$ is an abelian Lie ring	$f(x) \cdot y = f(y) \cdot x \ \forall \ x,y \in L$
$\varphi$ is the zero map and $L$ is an abelian Lie ring	No additional condition