# KillingForm -- computes the scaled Killing form applied to two weights

## Synopsis

• Usage:
KillingForm(g,v,w)
• Inputs:

## 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 : LieAlgebra i2 : KillingForm(g,{1,0},{0,1}) 1 o2 = - 3 o2 : QQ

## Ways to use KillingForm :

• "KillingForm(LieAlgebra,List,List)"
• "KillingForm(String,ZZ,List,List)"

## For the programmer

The object KillingForm is .