Semiholonomic jets and induced modules in Cartan geometry calculus

Jan Slovák; Vladimír Souček

Archivum Mathematicum (2024)

  • Volume: 060, Issue: 4, page 191-219
  • ISSN: 0044-8753

Abstract

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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.

How to cite

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Slovák, Jan, and Souček, Vladimír. "Semiholonomic jets and induced modules in Cartan geometry calculus." Archivum Mathematicum 060.4 (2024): 191-219. <http://eudml.org/doc/299349>.

@article{Slovák2024,
abstract = {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.},
author = {Slovák, Jan, Souček, Vladimír},
journal = {Archivum Mathematicum},
keywords = {Cartan connections; BGG machinery; tractor calculus; induced modules; Verma modules; semiholonomic jets},
language = {eng},
number = {4},
pages = {191-219},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Semiholonomic jets and induced modules in Cartan geometry calculus},
url = {http://eudml.org/doc/299349},
volume = {060},
year = {2024},
}

TY - JOUR
AU - Slovák, Jan
AU - Souček, Vladimír
TI - Semiholonomic jets and induced modules in Cartan geometry calculus
JO - Archivum Mathematicum
PY - 2024
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 060
IS - 4
SP - 191
EP - 219
AB - 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.
LA - eng
KW - Cartan connections; BGG machinery; tractor calculus; induced modules; Verma modules; semiholonomic jets
UR - http://eudml.org/doc/299349
ER -

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