We remark that it would be interesting to describe all finite-dimensional n-ary algebras with the property QDer = End. The attempt of doing that in the case of Lie triple systems (n = 3) was made in the paper . Unfortunately, as was noted in  (Date of publication: 2/06/2015), authors made a mistake in the classification of irreducible End(T)-submodules of T ⊗ T ⊗ T with respect to the action , where T is a Lie triple system (Lemma 4.2). Particularly, it was proposed that the sets(1)
are the only two nontrivial End(T)-submodules with respect to the action defined in . However, it is easy to see that the space
is an End(T)-submodule with respect to the action above and that it does not coincide with any of the two spaces above. Moreover, as was noted in  (Date of publication: 12/08/2015), the submodule (1) has a nontrivial irreducible End(T)-submodule
and the submodule (2) has a nontrivial irreducible End(T)-submodule
Also we note that in the proof of the Theorem 4.3  the authors have not used the second defining identity of Lie triple systems:
Therefore, they actually claim to have classified 3-Lie algebras with the property QDer = End. But from the classification of 3-Lie algebras with End = QDer [1, 3], it follows that some 3-Lie algebras with the above property are missing from their list.
I. Kaygorodov was supported by RFBR 16-31-00096.
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About the article
Published Online: 2016-07-26
Published in Print: 2016-01-01
Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 543–544, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0049.
© 2016 Kaygorodov and Popov, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0