1-cocycle for a Lie ring action

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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

Related notions