The KF object is designed and created to represent the Killing form of a Lie algebra.

By definition, the Killing form K of a Lie algebra L is the symmetric bilinear form on L and defined by K(V,V') = tr(ad V. adV') where V,V' are vector fields in L.

The KF object can only be constructed via query the Killing form of a Lie algebra. That is, let L be a LAVF object then the call K := KillingForm(L) construct a KF object K. object. See KillingForm for more detail.

Some methods become available once a valid KF object is constructed. See below for more details.

After a KF object K is successfully constructed, each method of K can be accessed by either the short form method(K, arguments). Note that the long form K:-method(K, arguments) would not work because the KF object is designed as a  local Maple object.

The following is a list of available methods for a KF object.

The KF object can act as the symmetric bilinear operator. See KF Object as Operator for more detail.

$\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)\:$

$V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x\,y\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$V≔\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{E2}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$L≔\mathrm{LAVF}\left(V\,\mathrm{E2}\right)$

$L≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

$K≔\mathrm{KillingForm}\left(L\right)$

$K≔\left(X\,Y\right)\mapsto -2\cdot \left(\frac{\partial }{\partial y}X\left(x\right)\right)\cdot \left(\frac{\partial }{\partial y}Y\left(x\right)\right)$

$\mathrm{type}\left(K\,'\mathrm{object}'\right)$

$\mathrm{true}$

$\mathrm{type}\left(K\,'\mathrm{KF}'\right)$

Error, type `KF` does not exist

$X≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x\,y\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$X≔\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)$

$Y≔\mathrm{subs}\left(\left[\mathrm{\xi }=\mathrm{\alpha }\,\mathrm{\eta }=\mathrm{\beta }\right]\,X\right)$

$Y≔\mathrm{\alpha }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\beta }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆy}\right)$

$K\left(X\,Y\right)$

$-2\left({\mathrm{\xi }}_y\right)\left(\frac{\partial }{\partial y}\mathrm{\alpha }\left(x\,y\right)\right)$

$\mathrm{GetMatrix}\left(K\right)$

$\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& \mathrm{-2}\end{array}\right]$

$\mathrm{IsTrivial}\left(K\right)$

$\mathrm{false}$

$\mathrm{IsNondegenerate}\left(K\right)$

$\mathrm{false}$