Characteristic Lie subring not implies ideal
ANALOGY BREAKDOWN: This is the breakdown of the analogue in Lie rings of a fact encountered in group. The old fact is: characteristic implies normal.
View other analogue breakdowns of characteristic implies normal|View other analogue breakdowns from group to Lie ring
This article gives the statement and possibly, proof, of a non-implication relation between two Lie subring properties. That is, it states that every Lie subring satisfying the first Lie subring property (i.e., characteristic Lie subring) need not satisfy the second Lie subring property (i.e., ideal of a Lie ring)
View a complete list of Lie subring property non-implications | View a complete list of Lie subring property implications
Get more facts about characteristic Lie subring|Get more facts about ideal of a Lie ring
Analogues in other algebraic structures
Example of the simple Witt algebra
- The additive group has basis . Explicitly, it is .
- The Lie bracket is defined as follows on the basis:
Consider the "sandwich" Lie subring of given by:
- is clearly a subring of .
- is characteristic in , because (note that the set with this description is not always a subring, but in this case it is).
- is not an ideal in : for instance, is not in . In fact, is simple, so it has no proper nonzero ideal.