nLab
coderivation

Definition

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

ΔD=(Did+idD)Δ:CCC \Delta \circ D = (D\otimes id + id \otimes D)\circ\Delta : C\to C\otimes C

If the commutative ring kk is a field and CC is finite-dimensional, then the transposes of the coderivations on CC are the derivations of the kk-algebra Hom k(C,k)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)H=U(L) of a Lie algebra LL on itself L h(g)=hgL_h (g) = h g restricts to an action of LL on U(L)U(L) by coderivations:

Δ(h)=1h+h1,\Delta (h) = 1\otimes h + h \otimes 1,

hence

(L h1+1L h)(Δ(g))=Δ(h)Δ(g)=Δ(hg)=(ΔL h)(g).(L_h \otimes 1 + 1 \otimes L_h)(\Delta (g)) = \Delta(h)\cdot\Delta(g) = \Delta(hg) = (\Delta \circ L_h)(g).

Revised on July 30, 2009 18:38:58 by Toby Bartels (71.104.230.172)