# 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