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 $f\cdot x\otimes y\otimes z={f}^{\ast}(x\otimes y\otimes z)$, where *T* is a Lie triple system (Lemma 4.2). Particularly, it was proposed that the sets

and

$$(T\otimes T\otimes T{)}^{-}=span(x\otimes y\otimes z-y\otimes x\otimes z:x,y,z\in T)$$(2)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:

$$[x,y,z]+[y,z,x]+[z,x,y]=0.$$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

- [1]
Kaygorodov I., (

*n*+ 1)-ary derivations of semisimple Filippov algebras, Math. Notes, 96 (2014), 1-2, 206-216. [Web of Science] - [2]
Kaygorodov I., Popov Yu., Generalized derivations of (color)

*n*-ary algebras, Arxiv: 1506.00734 - [3]
Kaygorodov I., Popov Yu., Generalized derivations of (color)

*n*-ary algebras, Linear Multilinear Algebra, 64 (2016), 6, 1086-1106. - [4]
Zhou J., Chen L., Ma Y., Generalized derivations of Lie triple systems, Open Math., 14 (2016), 260-271

## About the article

**Received**: 2016-05-25

**Accepted**: 2016-07-04

**Published Online**: 2016-07-26

**Published in Print**: 2016-01-01

**Citation Information: **Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0049. Export Citation

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