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invariant scalar product

(Definition)

Let $\mathbb{K}$ be a field and a vector space over $\mathbb{K}$ . Let be a group with a specified representation on denoted by for $v \in V$ and $g \in G$ .

An invariant scalar product (with respect to the action of ) on is a scalar product $\left( \cdot \lvert \cdot \right)$ on (i.e. a non-degenerate, symmetric $\mathbb{K}$ -bilinear form) such that for any $g \in G, u,v \in V$ we have

$\displaystyle \left( g . u \lvert g . v \right) = \left( u \lvert v \right)$

Now let $\mathfrak{g}$ be a Lie algebra over $\mathbb{K}$ with a representation on denoted by for $X \in \mathfrak{g}, v \in V$ . Then an invariant scalar product (with respect to the action of $\mathfrak{g}$ ) is a scalar product on such that for any $X \in \mathfrak{g}, u,v \in V$ we have

$\displaystyle \left( X . u \lvert v \right) = - \left( u \lvert X . v \right)$

An invariant scalar product on a Lie algebra $\mathfrak{g}$ is by definition an invariant scalar product as above where the representation is the adjoint representation of $\mathfrak{g}$ on itself. In this case invariance is usualy written $\left( [X, Y] \mid Z \right) = \left( X \mid [Y, Z] \right)$

Examples

For example if the orthogonal subgroup of $n \times n$ real matricies and $\mathbb{R}^n$ is the natural representation for , then the standard Euclidean scalar product on $\mathbb{R}^n$ is an invariant scalar product. Invariance in this example follows from the definition of .

As another example if $\mathfrak{g}$ is a complex semi-simple Lie algebra then the Killing form $\kappa(X,Y) := Tr(ad_X \cdot ad_Y)$ is an invariant scalar product on $\mathfrak{g}$ itself via the adjoint representation. Invariance in this example follows from the fact that the trace operator is associative, i.e. $Tr([Y,X] \cdot Z) = - Tr([X,Y] \cdot Z) = - Tr(X \cdot [Y,Z])$ . Thus an invariant scalar product (with respect to a Lie algebra representation) is sometimes called an associative scalar product.

"invariant scalar product" is owned by benjaminfjones.

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See Also: dot product

Other names:

invariant bilinear form, associative bilinear form

Also defines:

invariant scalar product, associative bilinear form, Killing form

Keywords:

scalar product, group action, Killing form

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Cross-references: associative, operator, trace, semi-simple Lie algebra, complex, Euclidean, real, subgroup, orthogonal, adjoint representation, Lie algebra, symmetric, non-degenerate, scalar product, action, representation, group, vector space, field
There is 1 reference to this entry.

This is version 4 of invariant scalar product, born on 2005-09-09, modified 2006-01-19.
Object id is 7364, canonical name is InvariantScalarProduct.
Accessed 4803 times total.

Classification:

AMS MSC:	15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
	22E60 (Topological groups, Lie groups :: Lie groups :: Lie algebras of Lie groups)
	22E10 (Topological groups, Lie groups :: Lie groups :: General properties and structure of complex Lie groups)
	22E15 (Topological groups, Lie groups :: Lie groups :: General properties and structure of real Lie groups)
	22E20 (Topological groups, Lie groups :: Lie groups :: General properties and structure of other Lie groups)

Pending Errata and Addenda

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