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(ρ)

     Invariants(ρ, T)

Parameters

     ρ            - a representation ρ of a Lie algebra 𝔤 on a vector space V

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

 

Description

Examples

Description

• 

Let ρ : 𝔤  glV be a representation of a Lie algebra 𝔤  on a vector space V. A vector Y V  is an invariant vector for the representation ρ if ρxY = 0 for all x   𝔤.   

• 

Let W=TsrV be the vector space of type r, s tensors on V. Then the representation ρ : 𝔤  glV defines an induced representation ρ: 𝔤  glW.

• 

The procedure Invariants(ρ) 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(ρ, T) returns a basis for the subspace of tensors which belong to T and which are invariant with respect to the representation ρ .

Examples

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

 

Example 1.

We define a 6-dimensional representation of sl2and find the invariant vectors.

L := LieAlgebraData([[x1, x2] = - 2*x1, [x1, x3] = x2, [x2, x3] = - 2*x3], [x1, x2, x3], sl2);

L:=e1,e2=2e1,e1,e3=e2,e2,e3=2e3

(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);

ρ1:=e1,000000200000010000030000003100000020,e2,400000020000000000000000000020000004,e3,020000003100000010000030000002000000

(2.2)
sl2 > 

Inv := Invariants(rho1);

Inv:=13D_z3+D_z4

(2.3)

 

We check this result using the ApplyRepresentation command.

W1 > 

map2(ApplyRepresentation, rho1, [e1, e2, e3], Inv[1]);

0D_z1,0D_z1,0D_z1

(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:=e1,e3=e1,e2,e3=e1+e2

(2.5)
V > 

DGsetup(L):

Alg1 > 

DGsetup([x, y, z], V):

V > 

rho2 := Representation(Alg1, V, Adjoint(Alg1));

ρ2:=e1,001000000,e2,001001000,e3,110010000

(2.6)

 

There are no vector invariants.

Alg1 > 

F := [D_x, D_y, D_z];

F:=D_x,D_y,D_z

(2.7)
V > 

Invariants(rho2, F);

(2.8)

 

There is one 1-form invariant.

Alg1 > 

Omega := [dx, dy, dz];

Ω:=dx,dy,dz

(2.9)
V > 

Invariants(rho2, Omega);

dz

(2.10)

 

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

V > 

T1 := Tensor:-GenerateTensors([[dx, dy, dz], [D_x, D_y, D_z]]);

T1:=dxD_x,dxD_y,dxD_z,dyD_x,dyD_y,dyD_z,dzD_x,dzD_y,dzD_z

(2.11)
V > 

Inv1 := Invariants(rho2, T1);

Inv1:=dxD_x+dyD_y+dzD_z

(2.12)

 

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

V > 

T2 := Tensor:-GenerateSymmetricTensors([dx, dy, dz], 2);

T2:=dxdx,12dxdy+12dydx,12dxdz+12dzdx,dydy,12dydz+12dzdy,dzdz

(2.13)
V > 

Inv2 := Invariants(rho2, T2);

Inv2:=dzdz

(2.14)

 

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

V > 

T3 := Tensor:-GenerateTensors([T1, [dx, dy, dz]]);

T3:=dxD_xdx,dxD_xdy,dxD_xdz,dxD_ydx,dxD_ydy,dxD_ydz,dxD_zdx,dxD_zdy,dxD_zdz,dyD_xdx,dyD_xdy,dyD_xdz,dyD_ydx,dyD_ydy,dyD_ydz,dyD_zdx,dyD_zdy,dyD_zdz,dzD_xdx,dzD_xdy,dzD_xdz,dzD_ydx,dzD_ydy,dzD_ydz,dzD_zdx,dzD_zdy,dzD_zdz

(2.15)
V > 

Inv3 := Invariants(rho2, T3);

Inv3:=dxD_xdz+dyD_ydz+dzD_zdz,dxD_xdzdyD_ydz+dzD_xdx+dzD_ydy,dyD_xdz+dzD_xdy

(2.16)

 

We can check the validity of the these calculations in two steps. First we use the matrices for the representation ρ1  to construct linear vector fields on the representation space V. This gives a vector field realization Γ 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 Γ vanishes.

V > 

A := map2(ApplyRepresentation, rho2,[e1, e2,e3]);

A:=001000000,001001000,110010000

(2.17)
Alg1 > 

ChangeFrame(V);

Alg1

(2.18)
V > 

Gamma := map(convert, A, DGvector);

Γ:=zD_x,zD_x+zD_y,x+yD_xyD_y

(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]));

0dxD_x0dxD_x0dxD_x

(2.20)
V > 

Matrix(1, 3, (i,j) -> LieDerivative(Gamma[j], Inv2[i]));

0dxdx0dxdx0dxdx

(2.21)
V > 

Matrix(3, 3, (i,j) -> LieDerivative(Gamma[j], Inv3[i]));

0dxD_xdx0dxD_xdx0dxD_xdx0dxD_xdx0dxD_xdx0dxD_xdx0dxD_xdx0dxD_xdx0dxD_xdx

(2.22)

See Also

DifferentialGeometry

Tensor

Library

LieAlgebras

ApplyRepresentation

GenerateTensors

GenerateSymmetricTensors

Representation

Retrieve