Lazard-divided Lie ring

Definition

A Lazard-divided Lie ring is a Lie ring $L$ equipped with additional multilinear operations, one for each prime number $p$ , of the form:

$t_{p} : L \times L \times \dots \times L \to L$

where there are $p$ copies of $L$

such that the following holds for all $x_{1}, x_{2}, \dots, x_{p} \in L$ :

$p t_{p} (x_{1}, x_{2}, \dots, x_{p}) = [[\dots [x_{1}, x_{2}], \dots, x_{p}]$

and further, such that every identity for which some multiple is an identity in Lie ring theory must hold.

The operations $t_{p}$ are called the Lazard division operations.

More abstractly, a Lazard-divided Lie ring is a representation of the Lazard-divided Lie operad.