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Tensor[InvariantTensorsAtAPoint] - find tensors or differential forms which are invariant under the infinitesimal action of a set of matrices

Calling Sequences

      InvariantTensorsAtAPoint(A, S, options)

Parameters

     A        - a list of square matrices, with dimension equal to the dimension of the space on which the tensors S are defined

     S        - a list of tensors or differential forms, each of the same index type

     options  - the keyword argument output

 

Description

Examples

Description

• 

This command calculates the tensors in the span of the tensors in the list S which are invariant with respect to the infinitesimal action generated by the matrices in the list A. This is a pointwise calculation.

• 

 Let x1, x2, ... xn  be the coordinates in terms of which the tensors in the list S are defined. If P = pji and X= Xi i , then PX=  pj i Xj i  . If  α = ai dx i , then Pα= pj i aj dxi .  If  T1 and T2 are tensors, then PT1  T2 = PT1  T2 + T1 P T2. Thus, the action of P on a tensor T  defined at a point coincides with the Lie derivative of  T  (as a tensor with constant coefficients) with respect to the linear vector field ZP =  pj i xj i ,that is, P T = LZPT. See Example 6 for examples of this action of matrices on tensors.

• 

 If A = P1 , P2 , ... , Pn and S = T1 , T2 , ... , Tm, then InvariantTensorsAtAPoint(A, S) returns a basis for the vector space of all tensors T = t1T1  + t2T2  + .. . + tmTm (ti  constant) such that  P1T = P2T =. . .= PmT = 0.

• 

If no invariant tensors exist, an empty list is returned.

• 

With output = "list", a list of invariant tensors is returned. This is the default. With output = "general", a single tensor with arbitrary coefficients _C1 , _C2 , ... is returned. If the number of matrices in the list A is 1 and output = "action", then the action of the matrix in A on the tensors in S is returned.

• 

 In many cases, the list of tensors S to be used by InvariantTensorsAtAPoint can be created with the commands GenerateTensors, GenerateSymmetricTensors, GenerateForms.

Examples

withDifferentialGeometry:withTensor:withLieAlgebras:withGroupActions:

 

Example 1.

Define a list of matrices for the first argument of InvariantTensorsAtAPoint .

 

AMatrix1,0,0,1,Matrix0,1,0,0

A:=1001,0100

(2.1)

 

Define a 2-dimensional space on which the tensors S for the second argument of InvariantTensorsAtAPoint will be defined.

DGsetupx,y,M

frame name: M

(2.2)

 

We take for S the space of all rank 2 covariant tensors on M.

M > 

SevalDGdx &t dx,dx &t dy,dy &t dx,dy &t dy

S:=dxdx,dxdy,dydx,dydy

(2.3)
M > 

InvariantTensorsAtAPointA,S

dxdy+dxdy

(2.4)

 

Example 2.

 

Here we consider a simple example where the matrices A depend upon the coordinates of the manifold on which the tensors S are defined.

M > 

DGsetupx,y,z,M

frame name: M

(2.5)

AMatrix0,1,0,1,0,0,0,0,0,Matrix0,0,1,0,0,0,1y2,0,0,Matrix0,0,0,0,0,1,0,1y2,0

A:=010100000,0010001y200,00000101y20

(2.6)

 

We take for S the space of all symmetric rank-2 covariant tensors on M.

M > 

SevalDGdx &t dx,dx &s dy,dx &t dz,dy &t dy,dy &s dz,dz &t dz

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

(2.7)

 

We find that the A-invariant tensors vary with the coordinate y.

M > 

InvariantTensorsAtAPointA,S

dxdxy2+dydyy2+dzdz

(2.8)

 

Example 3.

The classical simple Lie algebras can be defined as matrix algebras which leave a tensor or a collection of tensors invariant. In this example we check that the 4 ×4 matrices defining the real sympletic algebra leave invariant a non-degenerate 2-form.

We first use the commands SimpleLieAlgebraData and StandardRepresentation to obtain the matrices defining sp4, R.

 

LDSimpleLieAlgebraDatasp(4, R),sp4R

LD:=e1,e2=e2,e1,e3=e3,e1,e5=2e5,e1,e6=e6,e1,e8=2e8,e1