ApplyHomomorphism(Phi, T) will apply the transformation Phi to the vector, form or tensor T and return an object of the same type.  The precise evaluation rules for ApplyHomomorphism depend upon the specific properties of T and whether or not Phi is invertible.  The details follow.

Applied to tensors, the command ApplyHomomorphism acts as a ring homomorphism, that is, ApplyHomomorphism(Phi, T &tensor S) = ApplyHomomorphism(Phi, T) &tensor ApplyHomomorphism(Phi, S).

CASE 1. T is a vector in the domain algebra g of Phi.  In this case ApplyHomomorphism(Phi, T) simply applies the linear transformation Phi to the vector T and the result is a vector in the range algebra k of the transformation Phi.

CASE 2. T is a k-form on the range algebra k of transformation Phi.  In this case ApplyHomomorphism(Phi, T) simply applies the pullback of the linear transformation Phi to the k-form T and the result is a k-form in the domain g of Phi.

CASE 3. T is a tensor on g and Phi is an invertible linear transformation.  Then ApplyHomomorphism(Phi, T) is the tensor on the range algebra k obtained by the pushforward by Phi of the contravariant components of T and the pullback of the covariant components of T by the inverse of Phi.

CASE 4. T is a tensor on k and Phi is an invertible linear transformation.  Then ApplyHomomorphism(Phi, T) is the tensor on the domain algebra g obtained by the pushforward of the contravariant components of T by the inverse of Phi and the pullback of the covariant components of T by Phi.

CASE 5. T is a tensor on g and Phi is not invertible.  Then T must be a contravariant tensor (that is, a tensor products of vectors) in which case ApplyHomomorphism(Phi, T) is the contravariant tensor defined on the range algebra k and obtained by the pushforward of Phi acting on vectors in g.

When k = g, Case 4 takes precedence over Case 5.  Alternatively ApplyHomomorphism can be forced to use Case 4 or Case 5 with the third optional argument "domain" or "range".

CASE 6. T is a tensor on k and Phi is not invertible.  Then T must be a covariant tensor (that is, a tensor product of 1-forms) in which case ApplyHomomorphism(Phi, T) is the covariant tensor defined on the domain algebra g and obtained by the pullback of Phi acting on 1-forms in k.

The command ApplyHomomorphism is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form ApplyHomomorphism(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-ApplyHomomorphism(...).