killingForm -- computes the scaled Killing form applied to two weights

Synopsis

Usage:

killingForm(g,v,w)
Inputs:
- g, an instance of the type LieAlgebra,
- v, a list,
- w, a list,

Description

Let $\mathbf{g}$ be a Lie algebra. The Killing form on $\mathbf{g}$ is the symmetric bilinear form given by $(x,y) = Tr(ad x ad y)$. It can restricted to a Cartan subalgebra $\mathbf{h}$ and transferred to $\mathbf{h}^*$, yielding a symmetric bilinear form on weights. One popular convention is to scale the Killing form so that $(\theta,\theta) =2$, where $\theta$ is the highest root.

i1 : g=simpleLieAlgebra("A",2)

o1 = g

o1 : simple LieAlgebra

i2 : killingForm(g,{1,0},{0,1})

     1
o2 = -
     3

o2 : QQ

Ways to use killingForm :

killingForm(LieAlgebra,List,List)
killingForm(LieAlgebra,Vector,Vector)

For the programmer

The object killingForm is a method function.