calculate the invariant vectors for a representation of a Lie algebra, calculate the invariant tensors for a tensor product representation of a Lie algebra - Maple Programming Help

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LieAlgebras[Invariants] - calculate the invariant vectors for a representation of a Lie algebra, calculate the invariant tensors for a tensor product representation of a Lie algebra

Calling Sequences

Invariants(${\mathbf{ρ}}$)

Invariants(${\mathbf{\rho }}$, T)

Parameters

- a representation of a Lie algebra on a vector space $V$

T         - list of tensors on $V$ defining a subspace of tensors invariant under the induced representation

Description

 • Let be a representation of a Lie algebra on a vector space $V$. A vector  is an invariant vector for the representation if for all .
 • Let be the vector space of type  tensors on Then the representation defines an induced representation .
 • The procedure Invariants(${\mathbf{\rho }}$) returns a basis for the vector subspace of invariant vectors for the representation rho. An empty list is returned if the zero vector is the only invariant vector.
 • The procedure Invariants(${\mathbf{\rho }}$, T) returns a basis for the subspace of tensors which belong to T and which are invariant with respect to the representation $\stackrel{‾}{\mathrm{\rho }}$ .

Examples

 > with(DifferentialGeometry): with(Tensor): with(Library): with(LieAlgebras):

Example 1.

We define a 6-dimensional representation of $\mathrm{sl}\left(2\right)$and find the invariant vectors.

 > L := LieAlgebraData([[x1, x2] = - 2*x1, [x1, x3] = x2, [x2, x3] = - 2*x3], [x1, x2, x3], sl2);
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}\right]$ (2.1)
 > DGsetup(L):
 sl2 > DGsetup([z1, z2, z3, z4, z5, z6], W1):
 W1 > M := [Matrix([[0, 0, 0, 0, 0, 0], [- 2, 0, 0, 0, 0, 0], [0, - 1, 0, 0, 0, 0], [0, - 3, 0, 0, 0, 0], [0, 0, - 3, - 1, 0, 0], [0, 0, 0, 0, - 2, 0]]), Matrix([[- 4, 0, 0, 0, 0, 0], [0, - 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 4]]), Matrix([[0, - 2, 0, 0, 0, 0], [0, 0, - 3, - 1, 0, 0], [0, 0, 0, 0, - 1, 0], [0, 0, 0, 0, - 3, 0], [0, 0, 0, 0, 0, - 2], [0, 0, 0, 0, 0, 0]])]:
 W1 > rho1 := Representation(sl2, W1, M);
 ${\mathrm{ρ1}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {0}& {0}& {0}\\ {-}{2}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {0}& {-}{3}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{3}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{2}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrrrr}{-}{4}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{2}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {2}& {0}\\ {0}& {0}& {0}& {0}& {0}& {4}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrrrr}{0}& {-}{2}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{3}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}& {-}{3}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{2}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.2)
 sl2 > Inv := Invariants(rho1);
 ${\mathrm{Inv}}{:=}\left[{-}\frac{{1}}{{3}}{}{\mathrm{D_z3}}{+}{\mathrm{D_z4}}\right]$ (2.3)

We check this result using the ApplyRepresentation command.

 W1 > map2(ApplyRepresentation, rho1, [e1, e2, e3], Inv[1]);
 $\left[{0}{}{\mathrm{D_z1}}{,}{0}{}{\mathrm{D_z1}}{,}{0}{}{\mathrm{D_z1}}\right]$ (2.4)

Example 2.

In this example we calculate the invariant (1,1) tensors, the invariant (0,2) symmetric tensors and the type (1,2) invariant tensors for the adjoint representation of the Lie algebra [3,2] in the Winternitz tables of Lie algebras. We begin by using the Retrieve command to obtain the the structure equations for this Lie algebra.

 W1 > L := Retrieve("Winternitz", 1, [3, 2], Alg1);
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}\right]$ (2.5)
 V > DGsetup(L):
 Alg1 > DGsetup([x, y, z], V):
 V > rho2 := Representation(Alg1, V, Adjoint(Alg1));
 ${\mathrm{ρ2}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrr}{-}{1}& {-}{1}& {0}\\ {0}& {-}{1}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.6)

There are no vector invariants.

 Alg1 > F := [D_x, D_y, D_z];
 ${F}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2.7)
 V > Invariants(rho2, F);
 $\left[{}\right]$ (2.8)

There is one 1-form invariant.

 Alg1 > Omega := [dx, dy, dz];
 ${\mathrm{Ω}}{:=}\left[{\mathrm{dx}}{,}{\mathrm{dy}}{,}{\mathrm{dz}}\right]$ (2.9)
 V > Invariants(rho2, Omega);
 $\left[{\mathrm{dz}}\right]$ (2.10)

There is 1 invariant type (1,1) tensor.

 V > T1 := Tensor:-GenerateTensors([[dx, dy, dz], [D_x, D_y, D_z]]);
 ${\mathrm{T1}}{:=}\left[{\mathrm{dx}}{}{\mathrm{D_x}}{,}{\mathrm{dx}}{}{\mathrm{D_y}}{,}{\mathrm{dx}}{}{\mathrm{D_z}}{,}{\mathrm{dy}}{}{\mathrm{D_x}}{,}{\mathrm{dy}}{}{\mathrm{D_y}}{,}{\mathrm{dy}}{}{\mathrm{D_z}}{,}{\mathrm{dz}}{}{\mathrm{D_x}}{,}{\mathrm{dz}}{}{\mathrm{D_y}}{,}{\mathrm{dz}}{}{\mathrm{D_z}}\right]$ (2.11)
 V > Inv1 := Invariants(rho2, T1);
 ${\mathrm{Inv1}}{:=}\left[{\mathrm{dx}}{}{\mathrm{D_x}}{+}{\mathrm{dy}}{}{\mathrm{D_y}}{+}{\mathrm{dz}}{}{\mathrm{D_z}}\right]$ (2.12)

There is 1 invariant symmetric type (0,2)  tensor (but no invariant metrics).

 V > T2 := Tensor:-GenerateSymmetricTensors([dx, dy, dz], 2);
 ${\mathrm{T2}}{:=}\left[{\mathrm{dx}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{dy}}{,}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.13)
 V > Inv2 := Invariants(rho2, T2);
 ${\mathrm{Inv2}}{:=}\left[{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.14)

There are 3 type (1,2) invariant tensors.

 V > T3 := Tensor:-GenerateTensors([T1, [dx, dy, dz]]);
 ${\mathrm{T3}}{:=}\left[{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{,}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{,}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{,}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{,}{\mathrm{dx}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{,}{\mathrm{dx}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{,}{\mathrm{dy}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{,}{\mathrm{dy}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{,}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{,}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{,}{\mathrm{dy}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{,}{\mathrm{dy}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{,}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{,}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{,}{\mathrm{dz}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{,}{\mathrm{dz}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{,}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{,}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dz}}\right]$ (2.15)
 V > Inv3 := Invariants(rho2, T3);
 ${\mathrm{Inv3}}{:=}\left[{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{+}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{+}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{,}{-}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{-}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{+}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{,}{-}{\mathrm{dy}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{+}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dy}}\right]$ (2.16)

We can check the validity of the these calculations in two steps. First we use the matrices for the representation  to construct linear vector fields on the representation space $V$. This gives a vector field realization of our Lie algebra. The invariance of the tensors Inv1, Inv2, Inv3 means that the Lie derivatives of these tensors with respect to the vector fields in $\mathrm{Γ}$ vanishes.

 V > A := map2(ApplyRepresentation, rho2,[e1, e2,e3]);
 ${A}{:=}\left[\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{-}{1}& {-}{1}& {0}\\ {0}& {-}{1}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]$ (2.17)
 Alg1 > ChangeFrame(V);
 ${\mathrm{Alg1}}$ (2.18)
 V > Gamma := map(convert, A, DGvector);
 ${\mathrm{Γ}}{:=}\left[{z}{}{\mathrm{D_x}}{,}{z}{}{\mathrm{D_x}}{+}{z}{}{\mathrm{D_y}}{,}{-}\left({x}{+}{y}\right){}{\mathrm{D_x}}{-}{y}{}{\mathrm{D_y}}\right]$ (2.19)

Use the LieDerivative command to verify the invariance of the the tensors calculated by the Invariants command.

 V > Matrix(1, 3, (i,j) -> LieDerivative(Gamma[j], Inv1[i]));
 $\left[\begin{array}{ccc}{0}{}{\mathrm{dx}}{}{\mathrm{D_x}}& {0}{}{\mathrm{dx}}{}{\mathrm{D_x}}& {0}{}{\mathrm{dx}}{}{\mathrm{D_x}}\end{array}\right]$ (2.20)
 V > Matrix(1, 3, (i,j) -> LieDerivative(Gamma[j], Inv2[i]));
 $\left[\begin{array}{ccc}{0}{}{\mathrm{dx}}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}{}{\mathrm{dx}}\end{array}\right]$ (2.21)
 V > Matrix(3, 3, (i,j) -> LieDerivative(Gamma[j], Inv3[i]));
 $\left[\begin{array}{ccc}{0}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}\\ {0}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}\\ {0}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}& {0}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}\end{array}\right]$ (2.22)