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

TL;DR

This paper investigates the conditions under which Euclidean metrics on graded Lie algebras are admissible for constructing normal Cartan connections, extending existing theories to non-semisimple and cotangent Lie algebras.

Contribution

It establishes necessary and sufficient conditions for admissible metrics and develops a theory of normal Cartan connections for non-semisimple graded Lie algebras.

Findings

Characterization of admissible Euclidean metrics for graded Lie algebras.

Extension of Cartan connection theory to non-semisimple and cotangent Lie algebras.

Integration of previous semisimple cases into a broader framework.

Abstract

Let \gh = \gh_{-k}\oplus \cdots \oplus \gh_{l} (k >0, l \geq 0) be a finite dimensional real graded Lie algebra, with a Euclidian metric \langle \cdot , \cdot \rangle adapted to the gradation. The metric \langle\cdot , \cdot \rangle is called admissible if the codifferentials \partial^{*} : C^{k+1}(\gh_{-}, \gh ) \ra C^{k} (\gh_{-}, \gh) (k\geq 0) are Ad_{Q}-invariant (Lie(Q) = \gh_{0}\oplus \gh_{+}). We find necessary and sufficient conditions for a Euclidian 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 by A. Cap and J. Slovak (Parabolic Geometry I, Mathematical Surveys and Monographs, vol. 154, 2009), about normal Cartan connections of semisimple type, fits into our theory. We also consider in some detail the case when \gh = t^{*} (\gg ) is the cotangent Lie algebra of a non-positively graded Lie algebra \gg.