Group schemes and rigidity of algebras in positive characteristic
It is well known that in positive (or finite) characteristic additional problems arise in the comparison of the infinitesimal and local points of view. For example trying to ``exponentiate'' the derivation \({\mathcal D}\) of a Lie algebra \(L\) to a formal automorphism \(\varphi=1+t{\mathcal D}+t^2\varphi_2+\cdots\), you meet a problem because the usual formula \(\varphi_n= \frac{1}{n!} {\mathcal D}^n\) stops working for \(n=p\), where \(p\) is the characteristic of the ground field. One can make a cocycle for \(H^2(L,L)\): \[ \psi(x,y)= \sum_{i=1,\dots,p-1} \frac{1}{i!(p-i)!} [{\mathcal D}^ix,{\mathcal D}^{p-i}y], \] which is denoted \(\text{Sq }{\mathcal D}\) and called obstruction. The author makes careful considerations of these problems and of the whole situation with deformation in this context. Following Gerstenhaber, he distinguishes geometric rigidity and formal analytic rigidity, provides several criteria for a finite-dimensional Lie or associative algebra \(L\) to be rigid in that sense, and discusses properties of the obstruction subspace in \(H^2(L,L)\). His Theorem 2 shows the special role of the automorphism scheme \(\Aut(L)\) in this context. As an application the scheme theoretic description of deformations of the Jacobson-Witt algebras \(W_n\) is given.
deformations of the Jacobson-Witt algebras
scientific article; zbMATH DE number 858617
scientific article