Nonexistence of invariant symmetric forms on generalized Jacobson-Witt algebras revisited

We provide a short alternative homological argument showing that any invariant symmetric bilinear form on simple modular generalized Jacobson-Witt algebras vanishes, and outline another, deformation-theoretic one, allowing to describe such forms on simple modular Lie algebras of contact type.

💡 Research Summary

The paper addresses the long‑standing problem of determining invariant symmetric bilinear forms on simple modular Lie algebras of Cartan type, focusing on the generalized Jacobson‑Witt algebras (W(n)) and the contact algebras (K(2r+1)). An invariant symmetric form on a Lie algebra (L) is precisely an element of the space ((S^{2}L^{})^{L}). Classical results identify this space with the first cohomology group (H^{1}(L, L^{})) where (L^{}) is regarded as the coadjoint module. The author exploits this identification to give a concise homological proof that ((S^{2}W(n)^{})^{W(n)}=0) for all (n\ge 1) and all characteristics (p>0).

The argument proceeds by endowing (W(n)) with its standard filtration by degree of differential operators. The associated graded algebra (\operatorname{gr}W(n)) is canonically isomorphic to (W(n)) itself, which allows the use of the Hochschild‑Serre spectral sequence for the extension \