Examples
- Any Lie algebra over a general ring instead of a field is an example of a Lie ring. Lie rings are not Lie groups under addition, despite the name.
- Any associative ring can be made into a Lie ring by defining a bracket operator .
- For an example of a Lie ring arising from the study of groups, let be a group with the commutator operation, and let be a central series in — that is the commutator subgroup is contained in for any . Then
- is a Lie ring with addition supplied by the group operation (which will be commutative in each homogeneous part), and the bracket operation given by
- extended linearly. Note that the centrality of the series ensures the commutator gives the bracket operation the appropriate Lie theoretic properties.
Read more about this topic: Lie Ring
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