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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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# Commentary to: Generalized derivations of Lie triple systems

Ivan Kaygorodov
• Corresponding author
• Universidade Federal do ABC, CMCC, Santo André, Brasil and Siberian State Aerospace University, Krasnoyarsk, Federation
• Email:
/ Yury Popov
• Novosibirsk State University, Novosibirsk, Federation
• Email:
Published Online: 2016-07-26 | DOI: https://doi.org/10.1515/math-2016-0049

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

$(T⊗T⊗T)+=span(x⊗y⊗z+y⊗x⊗z:x,y,z∈T)$(1)

and

$(T⊗T⊗T)−=span(x⊗y⊗z−y⊗x⊗z:x,y,z∈T)$(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

$span(x⊗y⊗z+x⊗z⊗y:x,y,z∈T)$

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

$(T⊗T⊗T)∗=span(∑σ∈S3xσ(1)⊗xσ(2)⊗xσ(3):x1,x2,x3∈T);$

and the submodule (2) has a nontrivial irreducible End(T)-submodule

$(T⊗T⊗T)∗∗=span(∑σ∈S3(−1)σxσ(1)⊗xσ(2)⊗xσ(3):x1,x2,x3∈T).$

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.

• [2]

Kaygorodov I., Popov Yu., Generalized derivations of (color) n-ary algebras, Arxiv: 1506.00734 Google Scholar

• [3]

Kaygorodov I., Popov Yu., Generalized derivations of (color) n-ary algebras, Linear Multilinear Algebra, 64 (2016), 6, 1086-1106. Google Scholar

• [4]

Zhou J., Chen L., Ma Y., Generalized derivations of Lie triple systems, Open Math., 14 (2016), 260-271 Google Scholar

Accepted: 2016-07-04

Published Online: 2016-07-26

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455,

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