DifferentialGeometry/Tensor/IsotropyType - Maple Help

Tensor[IsotropyType] - find the isotropy type for the infinitesimal isometries of a metric

Calling Sequences

IsotropyType(Gamma, pt, output)

IsotropyType(A, output)

IsotropyType(output)

Parameters

Gamma  - a list of vectors, the infinitesimal generators for the isometry group of a metric $g$, equivalently, the Killing vectors of $g$

pt     - a list of equations specifying the equations of a point

A      - a list of matrices defining a subalgebra of the Lorentz Lie algebra $\mathrm{so}\left(3,1\right)$

output - (optional) the keyword argument output = str, where str is one of "Notation", "Notation1", "Notation2",  "Notation3", "SO31I", "SO31II"

Description

 • Let $g$ be a metric on a 4 dimensional manifold $M$ with Lorentzian signature.  The isometry group of $g$ is the Lie group $G$ of transformations $\mathrm{φ}:M\to M$ which preserve the metric $g$ under pullback, that is, ${\mathrm{φ}}^{\ast }g=g$.  Pick a point $p\in M$.  Then the isotropy subgroup ${G}_{p}\subseteq G$ at $p$ is the subgroup of isometries $\mathrm{φ}\in G$ which fix the point $p$, that is, $\mathrm{φ}\left(p\right)=p$.  The Jacobian defines a representation of the isotropy subgroup ${G}_{p}$ as a subgroup of the Lorentz group $\mathrm{SO}\left(3,1\right)$.  This representation is called the isotropy type of ${G}_{p}$.
 • The command IsotropyType works at the infinitesimal level.  Let $\mathrm{Γ}$ be the infinitesimal isometry algebra of the metric $g$ and let ${\mathrm{Γ}}_{p}$ be the infinitesimal isotropy subalgebra of $\mathrm{Γ}$ at $p$. (${\mathrm{Γ}}_{p}$ is the Lie algebra of ${G}_{p}.)$  Let $\mathrm{so}\left(3,1\right)$ be the Lorentz Lie algebra, viewed as a set of linear transformations on ${T}_{p}M$.  The mapping $\mathrm{ρ}:{\mathrm{Γ}}_{p}\to \mathrm{so}\left(3,1\right)$ defined by  (where $\left[X,Y\right]$ is the Lie bracket of $X$ and $Y$), is called the infinitesimal linear isotropy representation of ${\mathrm{Γ}}_{p}$. It gives an identification of ${\mathrm{Γ}}_{p}$ with a subalgebra of the Lorentz Lie algebra $\mathrm{so}\left(3,1\right)$.
 • The subalgebras of $\mathrm{so}\left(3,1\right)$ have been classified (up to conjugation) by Patera and Winternitz and labeled as F1, F2, ..., F14. (Continuous subgroups of the fundamental groups of physics I. General method and the Poincare group, J. Math Physics, 16 (1975), 1597--1614).  Details of this classification are given in the examples below.
 • The command IsotropyType(Gamma, p) returns the Patera-Winternitz classification of the isotropy subalgebra ${\mathrm{Γ}}_{p}$.
 • The command KillingVectors can be used to calculate the infinitesimal isometry algebra $\mathrm{Γ}$ of the metric $g$.
 • The command IsotropySubalgebra can be used to calculate the infinitesimal isotropy subalgebra ${\mathrm{Γ}}_{p}$ and the linear infinitesimal isotropy representation.
 • The calling sequence IsotropyType(output = "Notation") returns a short description of the notation used by Patera/Winternitz.  The calling sequence IsotropyType(output = "SO31I") and IsotropyType(output = "SO31II") returns the explicit matrix basis used by Patera/Winternitz for the Lie algebra of the Lorentz group.
 • This command is part of the DifferentialGeometry:-Tensor package and so can be used in the form IsotropyType(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-IsotropyType.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$$\mathrm{with}\left(\mathrm{GroupActions}\right):$

Example 1.

We begin with two simple examples.  For the first example we use the metric (12.24a) from Stephani, Kramer et al.

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 M > $\mathrm{g1a}≔\mathrm{evalDG}\left(\frac{1\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}+\mathrm{dz}&t\mathrm{dz}\right)}{\left(4+{x}^{2}+{y}^{2}+{z}^{2}\right){4}^{-2}}-\mathrm{dt}&t\mathrm{dt}\right)$
 ${\mathrm{g1a}}{:=}{-}{\mathrm{dt}}{}{\mathrm{dt}}{+}\frac{{16}{}{\mathrm{dx}}{}{\mathrm{dx}}}{{4}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}}{+}\frac{{16}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{4}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}}{+}\frac{{16}{}{\mathrm{dz}}{}{\mathrm{dz}}}{{4}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}}$ (2.2)
 M > $\mathrm{Gamma1a}≔\mathrm{KillingVectors}\left(\mathrm{g1a},\mathrm{output}="list"\right)$
 ${\mathrm{Gamma1a}}{:=}\left[{-}\frac{{1}}{{16}}{}{z}{}{\mathrm{D_x}}{+}\frac{{1}}{{16}}{}{x}{}{\mathrm{D_z}}{,}{-}\frac{{1}}{{16}}{}{z}{}{\mathrm{D_y}}{+}\frac{{1}}{{16}}{}{y}{}{\mathrm{D_z}}{,}{-}\frac{{1}}{{16}}{}{y}{}{\mathrm{D_x}}{+}\frac{{1}}{{16}}{}{x}{}{\mathrm{D_y}}{,}{-}{\mathrm{D_t}}\right]$ (2.3)
 M > $\mathrm{IsotropyType}\left(\mathrm{Gamma1a},\left[x=0,t=0,y=0,z=0\right]\right)$
 ${"F3"}$ (2.4)
 M > $\mathrm{IsotropyType}\left(\mathrm{Gamma1a},\left[x=0,t=0,y=0,z=0\right],\mathrm{output}="Notation2"\right)$
 ${"\left[Rx, Ry, Rz\right]"}$ (2.5)

Example 2.

For our second example we use the metric (12.16) from Stephani, Kramer et al.

 M > $\mathrm{DGsetup}\left(\left[t,x,\mathrm{φ},\mathrm{θ}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.6)
 M > $\mathrm{g1b}≔\mathrm{evalDG}\left(\mathrm{dx}&t\mathrm{dx}-{\mathrm{sinh}\left(x\right)}^{2}\mathrm{dt}&t\mathrm{dt}+\mathrm{dtheta}&t\mathrm{dtheta}+{\mathrm{sin}\left(\mathrm{θ}\right)}^{2}\mathrm{dphi}&t\mathrm{dphi}\right)$
 ${\mathrm{g1b}}{:=}{-}{{\mathrm{sinh}}{}\left({x}\right)}^{{2}}{}{\mathrm{dt}}{}{\mathrm{dt}}{+}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}{+}{\mathrm{dtheta}}{}{\mathrm{dtheta}}$ (2.7)
 M > $\mathrm{Gamma1b}≔\mathrm{KillingVectors}\left(\mathrm{g1b}\right)$
 ${\mathrm{Gamma1b}}{:=}\left[\frac{{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{\mathrm{cos}}{}\left({\mathrm{φ}}\right){}{\mathrm{D_phi}}}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}{+}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{\mathrm{D_theta}}{,}{-}\frac{{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{\mathrm{D_phi}}}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}{+}{\mathrm{cos}}{}\left({\mathrm{φ}}\right){}{\mathrm{D_theta}}{,}{\mathrm{D_phi}}{,}{-}\frac{{{ⅇ}}^{{t}}{}{\mathrm{cosh}}{}\left({x}\right){}{\mathrm{D_t}}}{{\mathrm{sinh}}{}\left({x}\right)}{+}{{ⅇ}}^{{t}}{}{\mathrm{D_x}}{,}\frac{{{ⅇ}}^{{-}{t}}{}{\mathrm{cosh}}{}\left({x}\right){}{\mathrm{D_t}}}{{\mathrm{sinh}}{}\left({x}\right)}{+}{{ⅇ}}^{{-}{t}}{}{\mathrm{D_x}}{,}{-}{\mathrm{D_t}}\right]$ (2.8)
 M > $\mathrm{IsotropyType}\left(\mathrm{Gamma1b},\left[x=\mathrm{x0},t=\mathrm{t0},\mathrm{θ}=\mathrm{θ0},\mathrm{φ}=\mathrm{φ0}\right],\mathrm{output}="Notation2"\right)$
 ${"\left[Rz, Kz\right]"}$ (2.9)

Example 3.

In this example we explore some of the details regarding the classification of the subalgebras of the Lorentz algebra $\mathrm{so}\left(3,1\right)$.  With output = "SO31I" we obtain the standard basis for $\mathrm{so}\left(3,1\right)$ consisting of 3 rotations ${R}_{x},{R}_{y},{R}_{z}$ and 3 boosts ${K}_{x},{K}_{y},{K}_{z}$.

 M > $\mathrm{Rx},\mathrm{Ry},\mathrm{Rz},\mathrm{Kx},\mathrm{Ky},\mathrm{Kz}≔\mathrm{op}\left(\mathrm{IsotropyType}\left(\mathrm{output}="SO31I"\right)\right)$

With output = "SO31II" we obtain an alternative basis which is useful for listing the subalgebras.

 M > $\mathrm{B1},\mathrm{B2},\mathrm{B3},\mathrm{B4},\mathrm{B5},\mathrm{B6}≔\mathrm{op}\left(\mathrm{IsotropyType}\left(\mathrm{output}="SO31II"\right)\right)$

With output = "Notation" the relationship between these two basis is shown and the list of subalgebras of $\mathrm{so}\left(3,1\right)$, as given in the aforementioned paper of Patera and Winternitz, is listed:

 M > $\mathrm{IsotropyType}\left(\mathrm{output}="Notation"\right)$
 B1 = 2Rz, B2 = -2Kz, B3 = -Ry - Kz, B4 = Rx - Ky, B5 = Ry - Kx, B6 = Rx + Ky, B(theta) = cos(theta) Rz - sin(theta)Kz F1: {B1, B2, B3, B4, B5, B6} F2: {B1, B2, B3, B4} F3: {Rx, Ry, Rz} F4: {Rz, Kx, Ky} F5: {B(theta),B3, B4} F6: {B1, B3, B4} F7: {B2, B3, B4} F8: {B2 ,B3} F9: {B1, B2} F10: {B3, B4} F11: {B(theta)}) F12: {Rz} F13: {Kz} F14: {Ry + Kz} F15: {0}

As a simple consistency check on the IsotropyType program, let us pass to the program one of the matrix algebras from this list.

 M > $\mathrm{IsotropyType}\left(\left[\mathrm{B1},\mathrm{B3},\mathrm{B4}\right]\right)$
 ${"F6"}$ (2.10)

This classification result is independent of the basis used to define the isotropy algebra:

 M > $\mathrm{IsotropyType}\left(\left[\mathrm{B1}+\mathrm{B3},\mathrm{B3}-\mathrm{B4},\mathrm{B1}+2\mathrm{B4}\right]\right)$
 ${"F6"}$ (2.11)

This classification is independent of the basis used for the tangent space:

 M > $P≔\mathrm{Matrix}\left(\left[\left[1,1,0,1\right],\left[0,1,1,0\right],\left[1,0,0,1\right],\left[0,-1,0,1\right]\right]\right)$
 M > $\mathrm{C1},\mathrm{C3},\mathrm{C4}≔\mathrm{.}\left({P}^{-1},\mathrm{B1},P\right),\mathrm{.}\left({P}^{-1},\mathrm{B3},P\right),\mathrm{.}\left({P}^{-1},\mathrm{B4},P\right)$
 M > $\mathrm{IsotropyType}\left(\left[\mathrm{C1},\mathrm{C3},\mathrm{C4}\right]\right)$
 ${"F6"}$ (2.12)

Example 4

With infolevel[IsotropyType] := 2, the branching in the program can be followed.

 M > ${\mathrm{infolevel}}_{\mathrm{IsotropyType}}≔2$
 ${{\mathrm{infolevel}}}_{{\mathrm{DifferentialGeometry:-Tensor:-IsotropyType}}}{:=}{2}$ (2.13)
 M > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.14)
 M > $g≔\mathrm{evalDG}\left({y}^{3}\mathrm{dt}&t\mathrm{dt}-{y}^{3}\mathrm{dx}&t\mathrm{dx}-\mathrm{dy}&t\mathrm{dy}-{y}^{3}\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{:=}{{y}}^{{3}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}{{y}}^{{3}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{{y}}^{{3}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.15)
 M > $\mathrm{Gamma}≔\mathrm{KillingVectors}\left(g\right)$
 ${\mathrm{Γ}}{:=}\left[{z}{}{\mathrm{D_x}}{-}{x}{}{\mathrm{D_z}}{,}{-}{\mathrm{D_z}}{,}{-}{z}{}{\mathrm{D_t}}{-}{t}{}{\mathrm{D_z}}{,}{-}{x}{}{\mathrm{D_t}}{-}{t}{}{\mathrm{D_x}}{,}{-}{\mathrm{D_x}}{,}{\mathrm{D_t}}\right]$ (2.16)
 M > $\mathrm{IsotropyType}\left(\mathrm{Gamma},\left[t=4,x=2,y=2,z=1\right]\right)$
 Isotropy subalgebra has dimension 3 If isotropy subalgebra is 3 dimensional simple, then isotropy type is determined by the signature of the Killing form:    Matrix(3, 3, [[-3/2,1/4,0],[1/4,1/8,0],[0,0,2]])    If Killing form is negative-definite,  then isotropy type is "F3"    If Killing form is indefinite,  then isotropy type is "F4"    The command IsDefinite(h, 'query' = 'positive_definite') returns: false    The command IsDefinite(h, 'query' = 'indefinite') returns: true
 ${"F4"}$ (2.17)