Class two Lie cring

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Definition

A class two Lie cring is an abelian group $L$ (denoted additively) equipped with an additional binary operation $*$ satisfying the following additional conditions:

Condition name	Description
$*$ is a 2-cocycle for trivial group action from $L$ to itself	$(a*(b+c))+(b*c)=((a+b)*c)+(a*b)$ for all $a,b,c\in L$
$*$ is cyclicity-preserving	$a*b=0$ if $⟨a,b⟩$ is cyclic.
$*$ is skew-symmetric	$a*b=-(b*a)$ for all $a,b\in L$
$*$ has class two	$a*(b*c)=(a*b)*c=0$ for all $a,b,c\in L$ (note that because of skew symmetry, it suffices to assume that any one of the expressions is universally zero).

Equivalently, a class two Lie cring is a Lie cring satisfying the additional condition that $a*(b*c)=0$ for all $a,b,c$ in the Lie cring.

Facts

• Double of class two Lie cring is class two Lie ring

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Class two Lie ring

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Class two near-Lie cring