Derivation with divided Leibniz condition powers

Definition

Usual notion: derivation with divided Leibniz condition powers for all nonnegative integers

Suppose $R$ is a non-associative ring (i.e., a not necessarily associative ring). A derivation with divided Leibniz condition powers for $R$ is the following data:

For every nonnegative integer $i$, an endomorphism $d^{(i)}$ of the additive structure of $R$, with $d^{(0)}$ equal to the identity map. We denote $d^{(1)}$ by $d$.

satisfying the following Leibniz-type compatibility condition (divided version of binomial formula for powers of a derivation):

For all nonnegative integers $n$, we have $d^{(n)}(x*y)=\sum _{i+j=n}d^{(i)}(x)*d^{(j)}(y)$

Note that $d^{(1)}=d$ must be a derivation in the usual sense of the word.

Derivation with divided Leibniz condition powers up to a point

Suppose $R$ is a non-associative ring (i.e., a not necessarily associative ring) and $m$ is a natural number. A derivation with divided Leibniz condition powers up to $m$ for $R$ is the following data:

For every nonnegative integer $i\leq m$, an endomorphism $d^{(i)}$ of the additive structure of $R$, with $d^{(0)}$ equal to the identity map. We denote $d^{(1)}$ by $d$.

satisfying the following Leibniz-type compatibility condition (divided version of binomial formula for powers of a derivation):

For all nonnegative integers $n\leq m$, we have $d^{(n)}(x*y)=\sum _{i+j=n}d^{(i)}(x)*d^{(j)}(y)$

Note that $d^{(1)}=d$ must be a derivation in the usual sense of the word.

Facts

• Exponential of nilpotent derivation with divided Leibniz condition powers is endomorphism

Related notions

• Derivation with divided powers is a stronger condition that insists that the $d^{(n)}$ also form a system of divided powers of the derivation.