1-cocycle for a Lie ring action
Suppose is a Lie ring, is an abelian group, and is a Lie ring homomorphism from to the ring of endomorphisms of , where the latter gets the usual Lie ring structure from its structure as an associative ring.
A 1-cocycle for this action is a homomorphism of groups satisfying the additional condition:
If we suppress and denote the action by , this can be written as:
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 .
The set of 1-cocycles for the action of on 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|
|and is the adjoint action, i.e., is the inner derivation induced by||Also called derivation of a Lie ring|
|is the zero map||is the zero map on all of , i.e., it descends to a homomorphism from|
|is an abelian Lie ring|
|is the zero map and is an abelian Lie ring||No additional condition|