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

Authors : Atsushi NAKAJİMA

Pages : 103-124

Doi:10.24330/ieja.325932

View : 12 | Download : 11

Publication Date : 2017-07-11

Article Type : Research Paper

Abstract :Let $A$ be an associative algebra over a commutative ring $R$, $\text{BiL}insert ignore into journalissuearticles values(A);$ 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*} &\Omegainsert ignore into journalissuearticles values(A); = \{insert ignore into journalissuearticles values(f,\ \alpha);\ \vert \ f \in \text{Hom}_Rinsert ignore into journalissuearticles values(A,\ A);,\ \alpha \in \text{BiL}insert ignore into journalissuearticles values(A); \}, \\ &\text{TDer}insert ignore into journalissuearticles values(A); = \{insert ignore into journalissuearticles values(f,\ f`,\ f``); \in \text{Hom}_Rinsert ignore into journalissuearticles values(A,\ A);^3 \ \vert \ finsert ignore into journalissuearticles values(xy); = f`insert ignore into journalissuearticles values(x);y + xf``insert ignore into journalissuearticles values(y);\}. \end{align*} $\text{TDer}insert ignore into journalissuearticles values(A);$ is called the set of triple derivations of $A$. We define a Lie algebra structure on $\Omegainsert ignore into journalissuearticles values(A);$ and $\text{TDer}insert ignore into journalissuearticles values(A);$ such that $\varphi_A : \text{TDer}insert ignore into journalissuearticles values(A); \to \Omegainsert ignore into journalissuearticles values(A);$ is a Lie algebra homomorphism. \par Dually, for a coassociative $R$-coalgebra $C$, we define the $R$-modules $\Omegainsert ignore into journalissuearticles values(C);$ and $\text{TCoder}insert ignore into journalissuearticles values(C);$ which correspond to $\Omegainsert ignore into journalissuearticles values(A);$ and $\text{TDer}insert ignore into journalissuearticles values(A);$, and show that the similar results to the case of algebras hold. Moreover, since $C^* = \text{Hom}_Rinsert ignore into journalissuearticles values(C,\ R);$ is an associative $R$-algebra, we give that there exist anti-Lie algebra homomorphisms $\theta_0 : \text{TCoder}insert ignore into journalissuearticles values(C); \to \text{TDer}insert ignore into journalissuearticles values(C^*);$ and $\theta_1 : \Omegainsert ignore into journalissuearticles values(C); \to \Omegainsert ignore into journalissuearticles values(C^*);$ such that the following diagram is commutative : \begin{equation*} \begin{CD} \text{TCoder}insert ignore into journalissuearticles values(C); @>{\psi_C}>> \Omegainsert ignore into journalissuearticles values(C); \\ @VV{\theta_0}V  @VV{\theta_1} V  \\ \text{TDer}insert ignore into journalissuearticles values(C^*); @>{\varphi_{C^*}}>>\Omegainsert ignore into journalissuearticles values(C^*);. \end{CD} \end{equation*}
Keywords : Derivation, generalized derivation