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 |
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Also called derivation of a Lie ring |
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No additional condition |