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

Year 2017, , 103 - 124, 11.07.2017
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*}

References

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  • M. Bresar, On generalized biderivations and related maps, J. Algebra, 172(3) (1995), 764-786.
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  • 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.
Year 2017, , 103 - 124, 11.07.2017
https://doi.org/10.24330/ieja.325932

Abstract

References

  • 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.
There are 17 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Articles
Authors

Atsushi Nakajima This is me

Publication Date July 11, 2017
Published in Issue Year 2017

Cite

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. https://doi.org/10.24330/ieja.325932
AMA Nakajima A. ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS. IEJA. July 2017;22(22):103-124. doi:10.24330/ieja.325932
Chicago Nakajima, Atsushi. “ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS”. International Electronic Journal of Algebra 22, no. 22 (July 2017): 103-24. https://doi.org/10.24330/ieja.325932.
EndNote Nakajima A (July 1, 2017) ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS. International Electronic Journal of Algebra 22 22 103–124.
IEEE A. Nakajima, “ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS”, IEJA, vol. 22, no. 22, pp. 103–124, 2017, doi: 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 (July 2017), 103-124. https://doi.org/10.24330/ieja.325932.
JAMA Nakajima A. ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS. IEJA. 2017;22:103–124.
MLA Nakajima, Atsushi. “ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS”. International Electronic Journal of Algebra, vol. 22, no. 22, 2017, pp. 103-24, doi:10.24330/ieja.325932.
Vancouver Nakajima A. ON A LIE ALGEBRA RELATED TO SOME TYPES OF DERIVATIONS AND THEIR DUALS. IEJA. 2017;22(22):103-24.