ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS

Let $A$ be an associative algebra over a commutative ring $R$, $\text{BiL}(A)$ the set of $R$-bilinear maps from $A \times A$ to $A$, and arbitrarily elements $x$, $y$ in $A$. Consider the following $R$-modules: \begin{align*} &\Omega(A) = \{(f,\ \alpha)\ \vert \ f \in \text{Hom}_R(A,\ A),\ \alpha \in \text{BiL}(A) \}, \\ &\text{TDer}(A) = \{(f,\ f',\ f'') \in \text{Hom}_R(A,\ A)^3 \ \vert \ f(xy) = f'(x)y + xf''(y)\}. \end{align*} $\text{TDer}(A)$ is called the set of triple derivations of $A$. We define a Lie algebra structure on $\Omega(A)$ and $\text{TDer}(A)$ such that $\varphi_A : \text{TDer}(A) \to \Omega(A)$ is a Lie algebra homomorphism. \par Dually, for a coassociative $R$-coalgebra $C$, we define the $R$-modules $\Omega(C)$ and $\text{TCoder}(C)$ which correspond to $\Omega(A)$ and $\text{TDer}(A)$, and show that the similar results to the case of algebras hold. Moreover, since $C^* = \text{Hom}_R(C,\ R)$ is an associative $R$-algebra, we give that there exist anti-Lie algebra homomorphisms $\theta_0 : \text{TCoder}(C) \to \text{TDer}(C^*)$ and $\theta_1 : \Omega(C) \to \Omega(C^*)$ such that the following diagram is commutative : \begin{equation*} \begin{CD} \text{TCoder}(C) @>{\psi_C}>> \Omega(C) \\ @VV{\theta_0}V  @VV{\theta_1} V  \\ \text{TDer}(C^*) @>{\varphi_{C^*}}>>\Omega(C^*). \end{CD} \end{equation*}