A Mathematical Comment on Lanczos Potential Theory

The last invited lecture published in $1962$ by Lanczos on his potential theory is never quoted because it is in french. Comparing it with a commutative diagram in a recently published paper on gravitational waves, we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the formal theory of Lie pseudogroups through differential double duality and the construction of finite length differential sequences for Lie operators. We use the fact that a differential module $M$ defined by an operator ${\cal{D}}$ with coefficients in a differential field $K$ has vanishing first and second differential extension modules if and only if its adjoint differential module $N=ad(M)$ defined by the adjoint operator $ad({\cal{D}})$ is reflexive, that is $ad({\cal{D}})$ can be parametrized by the operator $ad({\cal{D}}_1)$ when ${\cal{D}}_1$ generates the compatibilty conditions (CC) of ${\cal{D}}$ while $ad({\cal{D}}_1)$ can be parametrized by $ad({\cal{D}}_2)$ when ${\cal{D}}_2$ generates the CC of ${\cal{D}}_1$. We provide an explicit description of the potentials allowing to parametrize the Riemann and the Weyl operators in arbitrary dimension, both with their respective adjoint operators.