weight (Lie algebras)
Let be an abelian Lie algebra, a vector space and
a representation. Then the representation
is said to be diagonalisable, if can be written as a direct
sum
where is the dual space of and
Now let be a semi-simple Lie algebra. Fix a Cartan subalgebra
, then is abelian. Let be a representation whose restriction to is
diagonalisable. Then for any , the space
is the weight space of with respect to
. The multiplicity of
with respect to is the dimension
of :
If the multiplicity of is greater than zero, then is called a weight of the representation .
A representation of a semi-simple Lie algebra is determined by the multiplicities of its weights.
Title | weight (Lie algebras) |
---|---|
Canonical name | WeightLieAlgebras |
Date of creation | 2013-03-22 13:11:42 |
Last modified on | 2013-03-22 13:11:42 |
Owner | GrafZahl (9234) |
Last modified by | GrafZahl (9234) |
Numerical id | 7 |
Author | GrafZahl (9234) |
Entry type | Definition |
Classification | msc 17B20 |
Synonym | weight |
Defines | diagonalisable |
Defines | diagonalizable |
Defines | multiplicity |
Defines | weight space |