Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON SOLVABILITY AND NILPOTENCY OF ALGEBRAS WITH BRACKET

  • Received : 2016.04.01
  • Published : 2017.03.01
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Abstract

We analyze properties of solvable and nilpotent algebras with bracket. The class of solvability and nilpotency of the tensor square of an algebra with bracket is obtained. Homological characterizations of nilpotent algebras with bracket are presented.

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References

  1. R. Baer, Representations of groups as quotient groups. I, Trans. Amer. Math. Soc. 58 (1945), 295-347.
  2. R. Baer, Representations of groups as quotient groups. II. Minimal central chains of a group, Trans. Amer. Math. Soc. 58 (1945), 348-389. https://doi.org/10.1090/S0002-9947-1945-0015107-1
  3. R. Baer, Representations of groups as quotient groups. III. Invariants of classes of related representations, Trans. Amer. Math. Soc. 58 (1945), 390-419. https://doi.org/10.1090/S0002-9947-1945-0015108-3
  4. F. Borceux, A survey of semi-abelian categories, in Galois theory, Hopf algebras and semi-abelian categories, 27-60, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004.
  5. F. Borceux and D. Bourn, Mal'cev, protomodular, homological and semi-abelian categories, Mathematics and its Applications vol. 566, Kluwer Academic Publihers, 2004.
  6. J. M. Casas, Homology with trivial coefficients and universal central extensions of algebras with bracket, Comm. Algebra 35 (2007), no. 8, 2431-2449. https://doi.org/10.1080/00927870701326049
  7. J. M. Casas and T. Datuashvili, Non-commutative Leibniz-Poisson algebras, Comm. Algebra 34 (2006), no. 7, 2507-2530. https://doi.org/10.1080/00927870600651091
  8. J. M. Casas, T. Datuashvili, and M. Ladra, Left-Right noncommutative Poisson algebras, Cent. Eur. J. Math. 12 (2014), no. 1, 57-78.
  9. J. M. Casas and T. Pirashvili, Algebras with bracket, Manuscripta Math. 119 (2006), no. 1, 1-15. https://doi.org/10.1007/s00229-005-0551-8
  10. T. Everaert and T. Van der Linden, Baer invariants in semi-abelian categories. I. General theory, Theory Appl. Categ. 12 (2004), no. 1, 1-33.
  11. A. Frohlich, Baer-invariants of algebras, Trans. Amer. Math. Soc. 109 (1963), 221-244. https://doi.org/10.1090/S0002-9947-1963-0158920-3
  12. M. Gel'fand and I. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl. 13 (1980), 248-262. https://doi.org/10.1007/BF01078363
  13. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Math. 9, Springer, 1972.
  14. S. A. Huq, Commutator, nilpotency, and solvability in categories, Quart. J. Math. Oxford Ser. 19 (1968), 363-389. https://doi.org/10.1093/qmath/19.1.363
  15. N. Jacobson, Lie algebras, Interscience Publishers, John Wiley & Sons, New York-London, 1962.
  16. I. V. Kanatchikov, On field theoretic generalizations of a Poisson algebra, Rep. Math. Phys. 40 (1997), no. 2, 225-234. https://doi.org/10.1016/S0034-4877(97)85919-8
  17. C. R. Leedham-Green and S. McKay, Baer-invariants, isologism, varietal laws and homology, Acta Math. 137 (1976), no, 1-2, 99-150. https://doi.org/10.1007/BF02392415
  18. J.-L. Loday, A. Frabetti, F. Chapoton, and F. Goichot, Dialgebras and related operads, Lect. Notes in Math. 1763, Springer, 2001.
  19. F. I. Michael, A note on the Five Lemma, Appl. Categ. Structures 21 (2013), no. 5, 441-448. https://doi.org/10.1007/s10485-011-9273-0
  20. I. S. Rakhimov, I. M. Rikhsiboev, and W. Basri, Complete list of low dimensional complex associative algebras, arXiv 0910.0932 (2009).