Commentary to: Generalized derivations of Lie triple systems

Ivan Kaygorodov; Yury Popov

doi:10.1515/math-2016-0049

Article Open Access

Commentary to: Generalized derivations of Lie triple systems

Ivan Kaygorodov and Yury Popov

Published/Copyright: July 26, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

Keywords: Generalized derivation; Lie triple system

MSC 2010: 17A42; 17B40

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⋅x⊗y⊗z=f∗(x⊗y⊗z), 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, 206-216.10.1134/S0001434614070220Search in Google Scholar

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

[3] Kaygorodov I., Popov Yu., Generalized derivations of (color) n-ary algebras, Linear Multilinear Algebra, 64 (2016), 6, 1086-1106.10.1080/03081087.2015.1072492Search in Google Scholar

[4] Zhou J., Chen L., Ma Y., Generalized derivations of Lie triple systems, Open Math., 14 (2016), 260-27110.1515/math-2016-0024Search in Google Scholar

Received: 2016-5-25

Accepted: 2016-7-4

Published Online: 2016-7-26

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.