Lie Ring - Examples

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.

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