invariant scalar product
Now let be a Lie algebra over with a representation on denoted by for . Then an invariant scalar product (with respect to the action of ) is a scalar product on such that for any we have
An invariant scalar product on a Lie algebra is by definition an invariant scalar product as above where the representation is the adjoint representation of on itself. In this case invariance is usualy written
For example if the orthogonal subgroup of real matricies and is the natural representation for , then the standard Euclidean scalar product on is an invariant scalar product. Invariance in this example follows from the definition of .
As another example if is a complex semi-simple Lie algebra then the Killing form is an invariant scalar product on itself via the adjoint representation. Invariance in this example follows from the fact that the trace operator is associative, i.e. . Thus an invariant scalar product (with respect to a Lie algebra representation) is sometimes called an associative scalar product.
|Title||invariant scalar product|
|Date of creation||2013-03-22 15:30:16|
|Last modified on||2013-03-22 15:30:16|
|Last modified by||benjaminfjones (879)|
|Synonym||invariant bilinear form|
|Synonym||associative bilinear form|
|Defines||invariant scalar product|
|Defines||associative bilinear form|