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 [4]. Unfortunately, as was noted in [2] (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
and
are the only two nontrivial End(T)-submodules with respect to the action defined in [4]. 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 [3] (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 [4] 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.
Acknowledgement
I. Kaygorodov was supported by RFBR 16-31-00096.
References
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