Lie ring arising as the double of a class two Lie cring
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This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.
View a complete list of properties of Lie rings
VIEW RELATED: Lie ring property implications | Lie ring property non-implications |Lie ring metaproperty satisfactions | Lie ring metaproperty dissatisfactions | Lie ring property satisfactions | Lie ring property dissatisfactions
Definition
A Lie ring arising as the double of a class two Lie cring is defined as a Lie ring such that there exists the structure of a class two Lie cring with the same additive group and underlying set , such that the Lie bracket of is the double of the cring operation, i.e., if the cring operation is , then:
![{\displaystyle [x,y]=2(x*y)\ \forall \ x,y\in L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e1d53f8e582be1316a7e510c37625232c1df0e3)
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| abelian Lie ring | ||||
| Baer Lie ring | ||||
| LCS-Baer Lie ring | ||||
| Lie ring whose bracket is the double of a Lie bracket giving nilpotency class two |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Lie ring arising as the skew of a class two near-Lie cring | ||||
| Lie ring of nilpotency class two | double of class two Lie cring is class two Lie ring |