4
$\begingroup$

Let $\mathcal{lie}_n$ be the free Lie algebra generated by $n$ elements $x_1,\ldots, x_n$. A derivation $u\in \text{Der}(\mathcal{lie}_n)$ is called tangential if there exist $a_i\in \mathcal{lie}_n, i=1\ldots n$ such that $u(x_i)=[x_i,a_i]$. The definition can be found in section 3 of Alekseev and Torossian's paper "The Kashiwara-Vergne conjecture and Drinfeld's associators" arXiv:0802.4300v1.

My question is: why it has the name "tangential derivation", is there any historical reason or geometric intuition of it?

$\endgroup$
1
  • 1
    $\begingroup$ (It is usually a good idea to provide a complete refence to a paper, not just its title) $\endgroup$ Jul 22 '12 at 4:58
4
$\begingroup$

You may be interested in these Bar Natan's lecture.

The geometric intuition is that the $n$th Lie algebra of tangential derivations can be realized into the algebra of tangential differential operators (not derivations !) on $\mathfrak g^n$ for any (say finite dimensional) Lie algebra $\mathfrak g$. If $G$ is a Lie group with Lie algebra $\mathfrak g$, then it acts on $\mathfrak g$ by the adjoint action, and the action of a tangential differential operators is a differentiation in a direction which is tangential to the orbits of the adjoint action, hence the name.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.