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

Year 2017, Volume: 22 Issue: 22, 103 - 124, 11.07.2017

Atsushi Nakajima

https://doi.org/10.24330/ieja.325932

Abstract

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*}

Keywords

Derivation, generalized derivation

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