invariant scalar product

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

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

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

Now let $\mathfrak{g}$ be a Lie algebra over $\mathbb{K}$ with a representation on $V$ denoted by $X . v$ 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 $V$ such that for any $X\in\mathfrak{g},u,v\in V$ we have

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

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 $\left([X,Y]\mid Z\right)=\left(X\mid[Y,Z]\right)$

1 Examples

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

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.

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