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 thefollowing $R$-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.\parDually, 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 similarresults to the case of algebras hold. Moreover, since $C^* =\text{Hom}_R(C,\ R)$ is an associative $R$-algebra, we give thatthere 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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- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI