invariant scalar product

Let 𝕂 be a field and V a vector spaceMathworldPlanetmath over 𝕂. Let G be a group with a specified representationPlanetmathPlanetmath on V denoted by g.v for v∈V and g∈G.

An invariant scalar product (with respect to the action of G) on V is a scalar productMathworldPlanetmath (⋅|⋅) on V (i.e. a non-degenerate, symmetricPlanetmathPlanetmath 𝕂-bilinear formPlanetmathPlanetmath) such that for any g∈G,u,v∈V we have


Now let đ”€ be a Lie algebraMathworldPlanetmath over 𝕂 with a representation on V denoted by X.v for Xâˆˆđ”€,v∈V. Then an invariant scalar product (with respect to the action of đ”€) is a scalar product on V such that for any Xâˆˆđ”€,u,v∈V we have


An invariant scalar product on a Lie algebra đ”€ is by definition an invariant scalar product as above where the representation is the adjoint representationMathworldPlanetmath of đ”€ on itself. In this case invariance is usualy written ([X,Y]∣Z)=(X∣[Y,Z])

1 Examples

For example if G=Oⁱ(n) the orthogonalMathworldPlanetmath subgroupMathworldPlanetmathPlanetmath of n×n real matricies and ℝn is the natural representation for Oⁱ(n), then the standard Euclidean scalar product on ℝn is an invariant scalar product. Invariance in this example follows from the definition of Oⁱ(n).

As another example if đ”€ is a complex semi-simple Lie algebra then the Killing formMathworldPlanetmath Îș⁹(X,Y):=T⁹r⁹(a⁹dX⋅a⁹dY) is an invariant scalar product on đ”€ itself via the adjoint representation. Invariance in this example follows from the fact that the trace operator is associative, i.e. T⁹r⁹([Y,X]⋅Z)=-T⁹r⁹([X,Y]⋅Z)=-T⁹r⁹(X⋅[Y,Z]). 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