index of a Lie algebra
Let be a Lie algebra over and its vector space dual. For let denote the stabilizer of with respect to the co-adjoint representation.
The index of is defined to be
Examples
-
1.
If is reductive then . Indeed, and are isomorphic as representations for and so the index is the minimal dimension among stabilizers of elements in . In particular the minimum is realized in the stabilizer of any regular element of . These elemtents have stabilizer dimension equal to the rank of .
-
2.
If then is called a Frobenius Lie algebra. This is equivalent to condition that the Kirillov form given by is non-singular for some . Another equivalent condition when is the Lie algebra of an algebraic group is that is Frobenius if and only if has an open orbit on .
Title | index of a Lie algebra |
---|---|
Canonical name | IndexOfALieAlgebra |
Date of creation | 2013-03-22 15:30:47 |
Last modified on | 2013-03-22 15:30:47 |
Owner | benjaminfjones (879) |
Last modified by | benjaminfjones (879) |
Numerical id | 6 |
Author | benjaminfjones (879) |
Entry type | Definition |
Classification | msc 17B05 |
Defines | index of a Lie algebra |
Defines | Frobenius Lie algebra |
Defines | Kirillov form |