Question

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?

Answer 1

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.