Semiholonomic jets and induced modules in Cartan geometry calculus

Slovák, Jan ; Souček, Vladimír

Archivum mathematicum, Tome 60 (2024) no. 4, pp. 191-219

Abstract (VO)

The famous Erlangen Programme was coined by Felix Klein in 1872 as an algebraic approach allowing to incorporate fixed symmetry groups as the core ingredient for geometric analysis, seeing the chosen symmetries as intrinsic invariance of all objects and tools. This idea was broadened essentially by Elie Cartan in the beginning of the last century, and we may consider (curved) geometries as modelled over certain (flat) Klein’s models. The aim of this short survey is to explain carefully the basic concepts and algebraic tools built over several recent decades. We focus on the direct link between the jets of sections of homogeneous bundles and the associated induced modules, allowing us to understand the overall structure of invariant linear differential operators in purely algebraic terms. This allows us to extend essential parts of the concepts and procedures to the curved cases.

Détail

DOI : 10.5817/AM2024-4-191

Classification : 17B10, 17B25, 22E47, 58J60
Keywords: Cartan connections; BGG machinery; tractor calculus; induced modules; Verma modules; semiholonomic jets

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