 Centralizer - Maple Help

Centraliser

calculate the centraliser of one LAVF object in another.

Normaliser

calculate the normaliser of one LAVF object in another. Calling Sequence Centraliser( M, L) Normaliser( L, U) Parameters

 M, L - LAVF objects where $M\subseteq L$ (see IsSubspace for more detail) U - (optional) a LAVF object where $L\subseteq U$ Description

 • Let M, L be LAVF objects which are Lie algebras (see IsLieAlgebra) and  $M\subseteq L$  . Then Centraliser(M, L) computes the centraliser of M in L, namely $\left\{x\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}\in \phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}L\phantom{\rule[-0.0ex]{1.5ex}{0.0ex}}|\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}\left[x,M\right]=0\right\}$, as a new LAVF object.
 • Centraliser(M, L) is equivalent to Transporter(L, M, T) where T is a trivial LAVF (i.e. its determining system has trivial solutions) associated with vector fields of L. See the method Transporter for more detail.
 • The name Centralizer is provided as alias.
 • Similarly, let L, U be LAVF objects that are Lie algebras and $L\subseteq U$. Then Normaliser(L,U) computes the normaliser of L in U, namely $\left\{x\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}\in \phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}U\phantom{\rule[-0.0ex]{1.5ex}{0.0ex}}|\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}\left[x,L\right]\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}\in \phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}L\right\}$ , as a new LAVF object.
 • The second input argument U defaults to a universal LAVF associated with vector fields of L. That is, Normaliser(L) is equivalent to Normaliser(L, U) where U := LAVF(GetVectorField(L), "universal").
 • The call Normaliser(L,U) is equivalent to Transporter(U, L, L).
 • These methods are associated with the LAVF object. For more detail, see Overview of the LAVF object. 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)

Example 1:

 > $\mathrm{T2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${\mathrm{T2}}{≔}\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (3)

We first construct two LAVF objects

 > $\mathrm{LT2}≔\mathrm{LAVF}\left(V,\mathrm{T2}\right)$
 ${\mathrm{LT2}}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (4)
 > $\mathrm{LE2}≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${\mathrm{LE2}}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (5)

Both LAVFs are Lie algebras and L is indeed a subalgebra of LE2

 > $\mathrm{IsLieAlgebra}\left(\mathrm{LT2}\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsLieAlgebra}\left(\mathrm{LE2}\right)$
 ${\mathrm{true}}$ (7)

2-dim translation group is subspace of 2-dim Euclidean group

 > $\mathrm{IsSubspace}\left(\mathrm{LT2},\mathrm{LE2}\right)$
 ${\mathrm{true}}$ (8)

The centraliser of translations in E(2) is the translations themselves...

 > $\mathrm{Centraliser}\left(\mathrm{LT2},\mathrm{LE2}\right)$
 $\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (9)

Normaliser of translations in E(2) is E(2)

 > $\mathrm{Normaliser}\left(\mathrm{LT2},\mathrm{LE2}\right)$
 $\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (10)

Normaliser of E(2) in 'Lie algebra' of all vector fields is 4-dim, and includes E(2) as well as the uniform scalings...

 > $\mathrm{N2}≔\mathrm{Normaliser}\left(\mathrm{LE2}\right)$
 ${\mathrm{N2}}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{{\mathrm{\eta }}}_{{y}}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}\right]\right\}$ (11)
 > $\mathrm{SolutionDimension}\left(\mathrm{N2}\right)$
 ${4}$ (12)

Example 2:

 > $S≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x,x\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)=0,\mathrm{\eta }\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${S}{≔}\left[{{\mathrm{\xi }}}_{{x}{,}{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{\mathrm{\eta }}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (13)

L is an affine group on the line (i.e. acts on x only)...

 > $L≔\mathrm{LAVF}\left(V,S\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{x}{,}{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{\mathrm{\eta }}{=}{0}\right]\right\}$ (14)
 > $\mathrm{IsLieAlgebra}\left(L\right)$
 ${\mathrm{true}}$ (15)
 > $\mathrm{SolutionDimension}\left(L\right)$
 ${2}$ (16)

Normaliser of L in Lie algebra of all vector fields is an infinite Lie pseudogroup

 > $N≔\mathrm{Normaliser}\left(L\right)$
 ${N}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{x}{,}{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{0}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}\right]\right\}$ (17)
 > $\mathrm{SolutionDimension}\left(N\right)$
 ${\mathrm{\infty }}$ (18) Compatibility

 • The Centraliser and Normaliser commands were introduced in Maple 2020.