# 1-cocycle for a Lie ring action

From Groupprops

## Contents

## Definition

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.

### Explicit definition

A 1-cocycle for this action is a homomorphism of groups satisfying the additional condition:

Equivalently:

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 .

### Group structure

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 |