find tensors or differential forms which are invariant under the infinitesimal action of a set of matrices - Maple Programming Help

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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)


     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






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.




Example 1.

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






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


frame name: M



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


M > 





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 > 


frame name: M






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




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

M > 





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