nLab
coderivation

Definition

Given a (counital coassociative) $k$-coalgebra $(C,\Delta ,ϵ)$, a $k$-linear coderivation is a $k$-module map $D:C\to C$ satisfying the co-Leibniz rule

$$\Delta \circ D=(D\otimes \mathrm{id}+\mathrm{id}\otimes D)\circ \Delta :C\to C\otimes C$$\Delta \circ D = (D\otimes id + id \otimes D)\circ\Delta : C\to C\otimes C

If the commutative ring $k$ is a field and $C$ is finite-dimensional, then the transposes of the coderivations on $C$ are the derivations of the $k$-algebra ${\mathrm{Hom}}_k(C,k)$. With proper care of continuity conditions, this correspondence generalizes to other (for example infinite-dimensional) contexts.

Examples

The left translations of the elements of the universal enveloping algebra $H=U(L)$ of a Lie algebra $L$ on itself $L_h(g)=hg$ restricts to an action of $L$ on $U(L)$ by coderivations:

$$\Delta (h)=1\otimes h+h\otimes 1,$$\Delta (h) = 1\otimes h + h \otimes 1,

hence

$$(L_h\otimes 1+1\otimes L_h)(\Delta (g))=\Delta (h)\cdot \Delta (g)=\Delta (\mathrm{hg})=(\Delta \circ L_h)(g).$$(L_h \otimes 1 + 1 \otimes L_h)(\Delta (g)) = \Delta(h)\cdot\Delta(g) = \Delta(hg) = (\Delta \circ L_h)(g).