Rank1Elements - Maple Help

LieAlgebras[Rank1Elements] - calculate the rank 1 matrices in the span of a given list of matrices

Calling Sequences

Rank1Elements(A, option)

Rank1Elements(R, option)

Rank1Elements(S,T,option)

Parameters

A        - a list of square matrices

R        -  a representation of a Lie algebra

S,T      -  lists of vectors in a Lie algebra, for each vector x in S and y in T, the Lie bracket [x, y] must belong to the subspace spanned by the vectors T

option   -  (optional) the keyword argument, output = "Equations"

Description

 • Let be a list of square matrices and let be their general linear combination. The command Rank1Elements calculates the values of the coefficients so that the rank of the matrix $A$ is 1.
 • For the second calling sequence the matrices are taken to be the matrices defining the representation of a Lie algebra. For the third calling sequence the matrices defined by the adjoints matrices of the elements of S, restricted to the subspace T, are used. For the second and third calling sequences, the command Rank1Elements will return a list of vectors in the Lie algebras.
 • The matrix will have rank 1 if all the sub-matrices have vanishing determinants. With the keyword argument output = "Equations", the list of these determinant values and the list of coefficients  is returned.
 • Rank 1 matrices play a special role in various prolongation theories for Lie algebras. See, for example, TanakaProlongation

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

Define a list of 3$×3$ matrices.

 alg1 > A := [Matrix([[1,0, 0], [0,1, 0], [0,0]]), Matrix([[0,1, 0], [0,0, 0], [0,0, 1]]), Matrix([[0, 0, 1], [0,0, 0], [0,0, 0]])];
 ${A}{:=}\left[\left[\begin{array}{rrr}{1}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {1}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {1}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]$ (1)

Find all rank 1 matrices in the span of the matrices $A$.

 A1 > Rank1Elements(A);
 $\left[\left[\begin{array}{ccc}{0}& {0}& {\mathrm{_t3}}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]$ (2)

Define a list of four $2×2$ matrices whose span is all $2×2$ matrices .

 A1 > A := [Matrix([[1,0],[0,0]]), Matrix([[0,1],[0,0]]), Matrix([[0,0], [1,0]]), Matrix([[0,0], [0,1]])];
 ${A}{:=}\left[\left[\begin{array}{rr}{1}& {0}\\ {0}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {0}\\ {1}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {0}\\ {0}& {1}\end{array}\right]\right]$ (3)

Find the general form of a 22 matrix with rank 1.

 A1 > Rank1Elements(A);
 $\left[\left[\begin{array}{cc}\frac{{\mathrm{_t2}}{}{\mathrm{_t3}}}{{\mathrm{_t4}}}& {\mathrm{_t2}}\\ {\mathrm{_t3}}& {\mathrm{_t4}}\end{array}\right]\right]$ (4)

Example 2.

Choose a Lie algebra from the database of Lie algebras and construct its adjoint representation.

 A1 > LD := Library:-Retrieve("Winternitz", 1, [5, 36], alg3);
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}\right]$ (5)
 > DGsetup(LD);
 ${\mathrm{Lie algebra: alg3}}$ (6)

Define a 5-dimensional space to serve as the representation space.

 alg3 > DGsetup([x1, x2, x3, x4, x5], V);
 ${\mathrm{frame name: V}}$ (7)

 alg3 > rho := Adjoint(alg3, representationspace = V);
 ${\mathrm{ρ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrrr}{0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {1}& {-}{1}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrrr}{0}& {-}{1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrrr}{-}{1}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e5}}{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (8)

Find the vectors in the Lie algebra whose adjoint matrices have rank 1.

 alg3 > Rank1Elements(rho);
 $\left[{\mathrm{_t1}}{}{\mathrm{e1}}\right]$ (9)

Example 3.

We use the Lie algebra defined in Example 2 to illustrate the third calling sequence. We calculate the vector in the span of  whose adjoint matrix, restricted to , has rank 1.



 alg > Rank1Elements([e1, e2, e3], [e1, e2, e3, e4]);
 $\left[{\mathrm{_t1}}{}{\mathrm{e1}}{+}{\mathrm{_t3}}{}{\mathrm{e3}}\right]$ (10)
 alg > A := Adjoint(_t1*e1+_t3*e3, [e1, e2, e3, e4]);
 ${A}{:=}\left[\begin{array}{cccc}{0}& {-}{\mathrm{_t3}}& {0}& {\mathrm{_t1}}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]$ (11)

The result can be checked with the LinearAlgebra Rank command.

 alg > LinearAlgebra:-Rank(A);
 ${1}$ (12)

The equations defining the rank 1 matrices are:

 alg4 > Rank1Elements([e1, e2, e3], [e1, e2, e3, e4], output = "Equations");
 $\left\{{0}{,}{{\mathrm{_t2}}}^{{2}}{,}{-}{\mathrm{_t2}}{}{\mathrm{_t3}}\right\}{,}\left[{\mathrm{_t1}}{,}{\mathrm{_t2}}{,}{\mathrm{_t3}}\right]$ (13)