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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,dy