Tanaka structures (non holonomic G-structures) and Cartan connections

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Authors

ALEKSEEVSKY Dmitry DAVID Liana Rodica

Year of publication 2015
Type Article in Periodical
Magazine / Source Journal of Geometry and Physics
MU Faculty or unit

Faculty of Science

Citation
Doi http://dx.doi.org/10.1016/j.geomphys.2015.01.018
Field General mathematics
Keywords Tanaka structures; (normal) Cartan connections; Parabolic geometry; (prolongation of) G-structures
Description Let h = h(-k) circle plus ... circle plus h(1) (k > 0, l >= 0) be a finite dimensional graded Lie algebra, with a Euclidean metric <., .> adapted to the gradation. The metric <., .> is called admissible if the codifferentials partial derivative*: Ck+1 (h(-), j) -> C-k(h(-), h) (k >= 0) are Q-invariant (Lie(Q) = h(0) circle plus h(+)). We find necessary and sufficient conditions for a Euclidean metric, adapted to the gradation, to be admissible, and we develop a theory of normal Cartan connections, when these conditions are satisfied. We show how the treatment from Cap and Slovak (2009), about normal Cartan connections of semisimple type, fits into our theory. We also consider in detail the case when h := t*(g) is the cotangent Lie algebra of a non-positively graded Lie algebra g. (C) 2015 Elsevier B.V. All rights reserved.
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