invariant scalar product
Let be a field and a vector space over . Let be a group with a specified representation on denoted by for and .
An invariant scalar product (with respect to the action of ) on is a scalar product on (i.e. a non-degenerate, symmetric -bilinear form) such that for any we have
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
1 Examples
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 |
Canonical name | InvariantScalarProduct |
Date of creation | 2013-03-22 15:30:16 |
Last modified on | 2013-03-22 15:30:16 |
Owner | benjaminfjones (879) |
Last modified by | benjaminfjones (879) |
Numerical id | 7 |
Author | benjaminfjones (879) |
Entry type | Definition |
Classification | msc 22E15 |
Classification | msc 22E10 |
Classification | msc 22E60 |
Classification | msc 15A63 |
Classification | msc 22E20 |
Synonym | invariant bilinear form |
Synonym | associative bilinear form |
Related topic | DotProduct |
Defines | invariant scalar product |
Defines | associative bilinear form |
Defines | Killing form |