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LieTypes :: KillingForm

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



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})

o2 = -

o2 : QQ

Ways to use KillingForm :

For the programmer

The object KillingForm is a method function.