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

Let h=h−k⊕⋯⊕hlh=h−k⊕⋯⊕hl (k>0k>0, l≥0l≥0) be a finite dimensional graded Lie algebra, with a Euclidean metric 〈⋅,⋅〉〈⋅,⋅〉 adapted to the gradation. The metric 〈⋅,⋅〉〈⋅,⋅〉 is called admissible if the codifferentials ∂∗:Ck+1(h−,h)→Ck(h−,h)∂∗:Ck+1(h−,h)→Ck(h−,h) (k≥0k≥0) are QQ-invariant (Lie(Q)=h0⊕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)h≔t∗(g) is the cotangent Lie algebra of a non-positively graded Lie algebra gg.