SymbolAlgebra - Maple Help

LieAlgebras[SymbolAlgebra] - find the symbol algebra for a distribution

Calling Sequences

SymbolAlgebra(D, pt, alg, option)

Parameters

D       - a list of vectors on a manifold $M$

pt      - a list of equations specifying a point on $M$

alg     - a name or string, the name to be assigned to the symbol algebra

option  - (optional keyword arguments) output = "WeakDerivedFlag" , order = "Increasing", order = "Decreasing",

Description

 • In differential geometry a distribution is a set of vector fields defined on a manifold $M.$ The sequence of distributions defined inductively by

= ${\mathrm{𝒟}}^{\left(1\right)}$= [+ ${\mathrm{𝒟}}^{\left(i+1\right)}$= [+

is called the weak derived flag of the distribution. These distributions satisfy and [${\mathrm{𝒟}}^{\left(j\right)}]\subset {\mathrm{𝒟}}^{\left(i+j\right)}$.The symbol algebra for the distribution $𝒟$ at the point is the graded nilpotent Lie algebra defined by

where , ... ,

The grading weight ofis $p.$

 • The command SymbolAlgebra((D, pt, alg) returns the structure equations for the symbol algebra of at the point specified by the second argument. These structure equations can be initialized with DGsetup. The command DGinfo can be used to view the grading of the symbol algebra.
 • With the keyword argument output = "WeakDerivedFlag", the structure equations for the symbol algebra and the vector fields defining the weak derived flag [${𝒟}^{\left(0\right)}$are returned.
 • The basis used to define the symbol algebra is given in terms of decreasing weights. For example, if with dim dim and dim then and . With the keyword argument order = "Increasing", the basis used to define the symbol algebra is given in terms of increasing weights so that, for example, if then and .

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

In this example we find the symbol algebra for the Hilbert-Cartan equation .

First create a 5-dimensional manifold.

 > DGsetup([x, y, z, y1, y2], M);
 ${\mathrm{frame name: M}}$ (1)

Define the rank 3 distribution for the Hilbert-Cartan equation $z''={\left(y''\right)}^{2}$.

 M > Delta := [D_x + y1*D_y + y2*D_y1 + y2^2*D_z, D_y2];
 ${\mathrm{Δ}}{:=}\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{+}{{\mathrm{y2}}}^{{2}}{}{\mathrm{D_z}}{,}{\mathrm{D_y2}}\right]$ (2)

Calculate the symbol algebra and initialize.

 M > LD := SymbolAlgebra(Delta, [x = 0, y = 0, y1 = 0, y2 = 0, z = 0], alg);
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e5}}\right]$ (3)
 M > DGsetup(LD);
 ${\mathrm{Lie algebra: alg}}$ (4)

Here is the multiplication table.

 M > MultiplicationTable("LieTable");
 $\left[\begin{array}{ccccccc}{}& {|}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {|}& {0}& {-}{\mathrm{e3}}& {\mathrm{e4}}& {0}& {0}\\ {\mathrm{e2}}& {|}& {\mathrm{e3}}& {0}& {\mathrm{e5}}& {0}& {0}\\ {\mathrm{e3}}& {|}& {-}{\mathrm{e4}}& {-}{\mathrm{e5}}& {0}& {0}& {0}\\ {\mathrm{e4}}& {|}& {0}& {0}& {0}& {0}& {0}\\ {\mathrm{e5}}& {|}& {0}& {0}& {0}& {0}& {0}\end{array}\right]$ (5)

The algebra is nilpotent

 alg > Query("Nilpotent");
 ${\mathrm{true}}$ (6)

Here is the grading of the algebra as a list and in table format.

 ${\mathrm{Gr}}{:=}\left[{-}{1}{,}{-}{1}{,}{-}{2}{,}{-}{3}{,}{-}{3}\right]$ (7)
 alg > convert(Gr, DGgrading, "table", [e1, e2, e3, e4, e5]);
 ${\mathrm{table}}\left(\left[{-}{3}{=}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{,}{-}{2}{=}\left[{\mathrm{e3}}\right]{,}{-}{1}{=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]\right]\right)$ (8)

Example 2.

We continue with the distribution given in Example 1. We use the keyword argument output = "WeakDerivedFlag" to display the weak derived flag used to calculate the symbol algebra.

 alg > SymbolAlgebra(Delta, [x = 0, y = 0, y1 = 0, y2 = 0, z = 0], alg, output = "WeakDerivedFlag");
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e5}}\right]{,}\left[\left[{\mathrm{D_x}}{+}{\mathrm{y1}}{}{\mathrm{D_y}}{+}{{\mathrm{y2}}}^{{2}}{}{\mathrm{D_z}}{+}{\mathrm{y2}}{}{\mathrm{D_y1}}{,}{\mathrm{D_y2}}\right]{,}\left[{-}{2}{}{\mathrm{y2}}{}{\mathrm{D_z}}{-}{\mathrm{D_y1}}\right]{,}\left[{\mathrm{D_y}}{,}{-}{2}{}{\mathrm{D_z}}\right]\right]$ (9)

We use the second calling sequence to give the symbol algebra in ascending grading order.

 M > LD2 := SymbolAlgebra(Delta, [x = 0, y = 0, y1 = 0, y2 = 0, z = 0], alg2, order = "Increasing");
 ${\mathrm{LD2}}{:=}\left[\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}\right]$ (10)
 M > DGsetup(LD2);
 ${\mathrm{Lie algebra: alg2}}$ (11)

Here is the multiplication table and the grading.

 $\left[\begin{array}{ccccccc}{}& {|}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {|}& {0}& {0}& {0}& {0}& {0}\\ {\mathrm{e2}}& {|}& {0}& {0}& {0}& {0}& {0}\\ {\mathrm{e3}}& {|}& {0}& {0}& {0}& {-}{\mathrm{e1}}& {-}{\mathrm{e2}}\\ {\mathrm{e4}}& {|}& {0}& {0}& {\mathrm{e1}}& {0}& {\mathrm{e3}}\\ {\mathrm{e5}}& {|}& {0}& {0}& {\mathrm{e2}}& {-}{\mathrm{e3}}& {0}\end{array}\right]{,}\left[{-}{3}{,}{-}{3}{,}{-}{2}{,}{-}{1}{,}{-}{1}\right]$ (12)

Example 3.

In this example we find the symbol algebra for the jet space ${J}^{2}\left({R}^{2},R\right)$. First create an 8 dimensional manifold.

 alg2 > DGsetup([x, y, z, p, q, r, s, t], M3);
 ${\mathrm{frame name: M3}}$ (13)

Define the rank 3 distribution for the canonical system on jet space.

 M3 > Delta3 := evalDG([D_x + p*D_z + r*D_p + s*D_q, D_y + q*D_z + s*D_p + t*D_q, D_r, D_s, D_t]);
 ${\mathrm{Δ3}}{:=}\left[{\mathrm{D_x}}{+}{p}{}{\mathrm{D_z}}{+}{r}{}{\mathrm{D_p}}{+}{s}{}{\mathrm{D_q}}{,}{\mathrm{D_y}}{+}{q}{}{\mathrm{D_z}}{+}{s}{}{\mathrm{D_p}}{+}{t}{}{\mathrm{D_q}}{,}{\mathrm{D_r}}{,}{\mathrm{D_s}}{,}{\mathrm{D_t}}\right]$ (14)

Calculate the symbol algebra and initialize it.

 M3 > LD3 := SymbolAlgebra(Delta3, [x = 0, y = 0, z = 0, p =0, q = 0, r = 0, s = 0, t = 0], alg3);
 ${\mathrm{LD3}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e8}}\right]$ (15)
 M > DGsetup(LD3);
 ${\mathrm{Lie algebra: alg3}}$ (16)

Here is the multiplication table and the grading.

 $\left[\begin{array}{cccccccccc}{}& {|}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}& {\mathrm{e6}}& {\mathrm{e7}}& {\mathrm{e8}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {|}& {0}& {0}& {0}& {-}{\mathrm{e6}}& {0}& {0}& {0}& {0}\\ {\mathrm{e2}}& {|}& {0}& {0}& {0}& {-}{\mathrm{e7}}& {-}{\mathrm{e6}}& {0}& {0}& {0}\\ {\mathrm{e3}}& {|}& {0}& {0}& {0}& {0}& {-}{\mathrm{e7}}& {0}& {0}& {0}\\ {\mathrm{e4}}& {|}& {\mathrm{e6}}& {\mathrm{e7}}& {0}& {0}& {0}& {\mathrm{e8}}& {0}& {0}\\ {\mathrm{e5}}& {|}& {0}& {\mathrm{e6}}& {\mathrm{e7}}& {0}& {0}& {0}& {\mathrm{e8}}& {0}\\ {\mathrm{e6}}& {|}& {0}& {0}& {0}& {-}{\mathrm{e8}}& {0}& {0}& {0}& {0}\\ {\mathrm{e7}}& {|}& {0}& {0}& {0}& {0}& {-}{\mathrm{e8}}& {0}& {0}& {0}\\ {\mathrm{e8}}& {|}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[{-}{1}{,}{-}{1}{,}{-}{1}{,}{-}{1}{,}{-}{1}{,}{-}{2}{,}{-}{2}{,}{-}{3}\right]$ (17)