Overview - Maple Help

Overview of the KF Object

Description

 • 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.

KF Object Methods

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right]\right):$
 > $V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y\right)\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)

Construct a vector fields system for E(2).

 > $L≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (3)

Find the KillingForm of L

 > $K≔\mathrm{KillingForm}\left(L\right)$
 ${K}{≔}\left({X}{,}{Y}\right){→}{-}{2}{}\left(\frac{{\partial }}{{\partial }{y}}{}{X}{}\left({x}\right)\right){}\left(\frac{{\partial }}{{\partial }{y}}{}{Y}{}\left({x}\right)\right)$ (4)

Although the Killing form K is an KF object, KF is a local Maple object and is not visible to public.

 > $\mathrm{type}\left(K,'\mathrm{object}'\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{type}\left(K,'\mathrm{KF}'\right)$

The KF object K can act as a symmetric bilinear operator on vector fields,

 > $X≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y\right)\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${X}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (6)
 > $Y≔\mathrm{subs}\left(\left[\mathrm{\xi }=\mathrm{\alpha },\mathrm{\eta }=\mathrm{\beta }\right],X\right)$
 ${Y}{≔}{\mathrm{\alpha }}{}\left({x}{,}{y}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\beta }}{}\left({x}{,}{y}\right){}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (7)
 > $K\left(X,Y\right)$
 ${-}{2}{}\left({{\mathrm{\xi }}}_{{y}}\right){}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\alpha }}{}\left({x}{,}{y}\right)\right)$ (8)

 > $\mathrm{GetMatrix}\left(K\right)$
 $\left[\begin{array}{ccc}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {-2}\end{array}\right]$ (9)
 > $\mathrm{IsTrivial}\left(K\right)$
 ${\mathrm{false}}$ (10)
 > $\mathrm{IsNondegenerate}\left(K\right)$
 ${\mathrm{false}}$ (11)