On smooth integration of p-nilpotent Lie algebras in positive characteristic

par Marion Jeannin

jeudi 11 déc. 2025, 14:00 → 16:00 Europe/Paris

M7-411

Let k be a field of characteristic zero, and let u be a nilpotent k-Lie algebra of finite dimension. The Baker–Campbell–Hausdorff formula, induced by the exponential map, defines a group law on the vector group V(u), making it into a unipotent algebraic k-group. In other
words, there is an equivalence between the category of nilpotent k-Lie algebras of finite dimension and unipotent algebraic k-groups. On the other hand, the functor G → Lie(G) induces a quasi-inverse equivalence. If now k is of characteristic p > 0, such a nice conversation between (unipotent algebraic) groups and (nilpotent) Lie algebras no longer exists in general, but one can still wonder whether under suitable assumptions it is still possible to associate a unipotent algebraic group to a “nilpotent” (this notion will need to be adapted to the context) Lie algebra. More precisely, in this talk we wonder whether, given a field of positive characteristic k, a reductive k-group G and a restricted p-nil subalgebra u of the Lie algebra of G, there exists
a smooth unipotent subgroup U ⊂ G such that Lie(U) = u. Obstructions are both arithmetic and algebraic : what will play the role of the exponential here? This will lead us to discuss assumptions to ensure the existence of Springer isomorphisms, and more specifically generalised
exponential maps; but also geometric: algebraic groups are no longer a priori smooth in positive
characteristic, a way of controlling the lack of smoothness is to refine the notion of infinitesimal
saturation, first introduced by Deligne.