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Open Mathematics

formerly Central European Journal of Mathematics

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

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IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

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Volume 14, Issue 1 (Jan 2016)

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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 fxyz=f(xyz), where T is a Lie triple system (Lemma 4.2). Particularly, it was proposed that the sets

(TTT)+=span(xyz+yxz:x,y,zT)(1)

and

(TTT)=span(xyzyxz:x,y,zT)(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(xyz+xzy:x,y,zT)

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

(TTT)=span(σS3xσ(1)xσ(2)xσ(3):x1,x2,x3T);

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

(TTT)=span(σS3(1)σxσ(1)xσ(2)xσ(3):x1,x2,x3T).

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 ScienceGoogle Scholar

  • [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

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.

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© 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

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