Geometric rigidity of modules for algebraic groups

Abstract: Let k be a field, let G be a smooth affine k-group of finite type, and V a finite-dimensional G-module. We say V is rigid if the socle series and radical series coincide for the action of G on each indecomposable summand of V; say V is geometrically rigid (resp. absolutely rigid) if V is rigid after base change of G and V to an algebraic closure of k (resp. any field extension of k). We show that all simple G-modules are geometrically rigid, though they are not in general absolutely rigid. More precisely, we show that if V is a simple G-module, then there is a finite purely inseparable extension k_V /k naturally attached to V such that V_{k_V} is absolutely rigid as a G_{k_V} -module. The proof for connected G turns on an investigation of algebras of the form K \otimes_k E where K and E are field extensions of k; we give an example of such an algebra which is not rigid as a module over itself. We establish the existence of the purely inseparable field extension k_V /k through an analogous version for Artinian algebras.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar