Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and $$\mathcal{L}_G:=\bigoplus_{k\geq 1}\mathcal{L}_G(k).$$ Then $\mathcal{L}_G$ has a graded Lie algebra structure induced from the commutator bracket on $G$.

I heard that $\mathcal{L}_G$ is called the Carnot Algebra. Could any one provide some reference of Carnot Algebras (definition, basic facts, etc.)? I never heard of this terminology; for all the papers that I have read, $\mathcal{L}_G$ is usually called the associated Lie algebra.

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    $\begingroup$ A Carnot (Lie) algebra is the Lie algebra of a certain simply connected nilpotent Lie group, namely of a Carnot group. $\endgroup$ – Dietrich Burde Feb 9 '15 at 14:08
  • $\begingroup$ Thanks @DietrichBurde! The wiki article seems quite concise; any introduction references available? $\endgroup$ – Zuriel Feb 9 '15 at 14:09
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    $\begingroup$ There is a good introduction of Carnot groups and algebras in Yves Cornuliers artilce here. $\endgroup$ – Dietrich Burde Feb 9 '15 at 14:11
  • $\begingroup$ Thanks @DietrichBurde for the link!! $\endgroup$ – Zuriel Feb 9 '15 at 14:18

This Algebra is called the Carnot group. Google and search for "Carnot Groups and Carnot solvmanifolds".

  • $\begingroup$ Many thanks for the reference! $\endgroup$ – Zuriel Feb 9 '15 at 14:25

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