Yıl 2017, Cilt 22, Sayı 22, Sayfalar 103 - 124 2017-07-11
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## ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS

#### Atsushi Nakajima [1]

##### 141 131

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*}

Derivation, generalized derivation
• M. Bresar, On the distance of the composition of two derivations to the gener- alized derivations, Glasgow Math. J., 33(1) (1991), 89-93.
• M. Bresar, On generalized biderivations and related maps, J. Algebra, 172(3) (1995), 764-786.
• C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and As- sociative Algebras, Pure and Applied Mathematics, Vol. XI, Interscience Pub- lishers, a division of John Wiley & Sons, New York-London, 1962.
• Y. Doi, Homological coalgebra, J. Math. Soc. Japan, 33(1) (1981), 31-50.
• M. Hongan and H. Komatsu, (sigma; tau)-derivations with invertible values, Bull. Inst. Math. Acad. Sinica, 15(4) (1987), 411-415.
• M. Hongan and H. Komatsu, On the module of differentials of a noncom- mutative algebra and symmetric biderivations of semiprime algebra, Comm. Algebra, 28(2) (2000), 669-692.
• H. Komatsu, Quasi-separable extensions of noncommutative rings, Comm. Al- gebra, 29(3) (2001), 1011-1019.
• H. Komatsu and A. Nakajima, Generalized derivations of associative algebras, Quaest. Math., 26(2) (2003), 213-235.
• H. Komatsu and A. Nakajima, On triple coderivations of corings, Int. Electron. J. Algebra, 17 (2015), 139-153.
• G. F. Leger and E. M. Luks, Generalized derivations of Lie algebras, J. Algebra, 228(1) (2000), 165-203.
• A. Nakajima, Coseparable coalgebras and coextensions of coderivations, Math. J. Okayama Univ., 22(2) (1980), 145-149.
• A. Nakajima, On categorical properties of generalized derivations, Sci. Math., 2(3) (1999), 345-352.
• A. Nakajima, Generalized Jordan derivations, International Symposium on Ring Theory (Kyongju, 1999), Trends Math., Birkhauser, Boston, MA, (2001), 235-243.
• A. Nakajima, Note on generalized Jordan derivations associate with Hochschild 2-cocycles of rings, Turkish J. Math., 30(4) (2006), 403-411.
• A. Nakajima, On generalized coderivations, Int. Electron. J. Algebra, 12 (2012), 37-52.
• M. E. Sweedler, Right derivations and right differential operators, Pacific J. Math., 86(1) (1980), 327-360.
• J. Vukman, Symmetric bi-derivations on prime and semi-prime rings, Aequa- tiones Math., 38(2-3) (1989), 245-254.
Konular Matematik Makaleler Yazar: Atsushi Nakajima
 Bibtex @araştırma makalesi { ieja325932, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Prof. Dr. Abdullah HARMANCI}, year = {2017}, volume = {22}, pages = {103 - 124}, doi = {10.24330/ieja.325932}, title = {ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS}, key = {cite}, author = {Nakajima, Atsushi} } APA Nakajima, A . (2017). ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS. International Electronic Journal of Algebra, 22 (22), 103-124. DOI: 10.24330/ieja.325932 MLA Nakajima, A . "ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS". International Electronic Journal of Algebra 22 (2017): 103-124 Chicago Nakajima, A . "ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS". International Electronic Journal of Algebra 22 (2017): 103-124 RIS TY - JOUR T1 - ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS AU - Atsushi Nakajima Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.325932 DO - 10.24330/ieja.325932 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 103 EP - 124 VL - 22 IS - 22 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.325932 UR - http://dx.doi.org/10.24330/ieja.325932 Y2 - 2018 ER - EndNote %0 International Electronic Journal of Algebra ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS %A Atsushi Nakajima %T ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 22 %N 22 %R doi: 10.24330/ieja.325932 %U 10.24330/ieja.325932 ISNAD Nakajima, Atsushi . "ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS". International Electronic Journal of Algebra 22 / 22 (Temmuz 2017): 103-124. http://dx.doi.org/10.24330/ieja.325932