Cohomology in algebra, geometry, physicsand statistics

Natural differential operators from Lie algebra homology

 I will present a generalization of construction of Calderbank and Diemer of the so called Bernstein-Gelfand-Gelfand sequences of differential operators which act on sections of bundles associated to any Cartan geometry of parabolic type. These operators were originally constructed by Čap, Slovák and Souček and include many important and interesting operators, e.g. operators whose kernels consists of various Killing fields or operators whose kernels provide Einstein metrics. Classically these operators are all strongly overdetermined but the generalization produces also other types of operators such as the conformally invariant modification of the Laplace-Beltrami operator or the Dirac operator. The construction of Calderbank and Diemer proceeds by explicitely constructing homotopy transfer data for a twisted deRham sequence. In the flat case (i.e. when the Cartan geometry is locally isomorphic to G/P) one obtains a complex of invariant differential operators that calculates the sheaf cohomology of a homogeneous vector bundle. On the algebraic side the crucial tool for the Calderbank-Diemer construction is Hodge theory for Chevalley-Eilenberg (co)homology of nilpotent Lie algebras with values in unitarizable representations.