Dynkin operator

Definition
Let $$R$$ be a commutative unital ring and $$S = R \langle T \rangle$$ be the free associative algebra over $$R$$ with freely generating set $$T$$. In other words, $$S$$ is the noncommutative polynomials with coefficients from $$R$$ and variables from $$T$$.

Left-normed Dynkin operator
The left-normed Dynkin operator, denoted $$\delta$$, is a $$R$$-linear mapping from $$S$$ to itself defined as follows:


 * On elements of $$R$$ and elements of $$T$$, it is the identity map.
 * It sends a monomial $$x_1x_2 \dots x_n$$ to the left-normed commutator expression $$[[\dots[[x_1,x_2],x_3],\dots,x_{n-1}],x_n]$$. Note that $$x_1,x_2,\dots,x_n$$ are possibly equal, possibly distinct elements of $$T$$. Note that $$n$$ is unrelated to the size of $$T$$.
 * The mapping is extended $$R$$-linearly to all polynomials.

We can think of the left-normed Dynkin operator as left multiplication by the Dynkin element in $$K[S_n]$$:

$$(1 - (1,2,\dots,n))(1 - (1,2,\dots,n-1)) \dots (1 - (1,2,3))(1 - (1,2))$$

Right-normed Dynkin operator
The right-normed Dynkin operator, denoted $$\delta$$, is a $$R$$-linear mapping from $$S$$ to itself defined as follows:


 * On elements of $$R$$ and elements of $$T$$, it is the identity map.
 * It sends a monomial $$x_1x_2 \dots x_n$$ to the right-normed commutator expression $$[x_1,[x_2,\dots,[x_{n-1},x_n]\dots ]]$$. Note that $$x_1,x_2,\dots,x_n$$ are possibly equal, possibly distinct elements of $$T$$.
 * The mapping is extended $$R$$-linearly to all polynomials.

Relation with Lie operad
The Dynkin operator describes the Lie operad. This essentially follows from Dynkin's lemma.