Higher Coderivations of formal triangular matrix Coalgebras

Abstract

In this paper, we introduce the notion of higher coderivations on coalgebras as an important generalization of coderivations, and show that there exists a one to one correspondence between the set of all higher coderivations $\{D_n\}$ with $D_0=id_C$ and the set of all sequences of coderivations $\{f_n\}$ with $f_0=0$. Then we characterize the higher coderivations on the formal triangular matrix coalgebra $\Gamma=\left(\begin{array}{cc} C & M\\ 0& D\\ \end{array}\right)$, and it is shown that the higher coderivations on $\Gamma$ could be described by the higher coderivations on $C$ and $D$, generalized higher coderivations on $M$, and two other related families of maps.