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Year 2020, , 187 - 192, 14.07.2020
https://doi.org/10.24330/ieja.768254

Abstract

References

  • D. W. Barnes, Some theorems on Leibniz algebras, Comm. Algebra, 39(7) (2011), 2463-2472.
  • C. Batten-Ray, A. Combs, N. Gin, A. Hedges, J. T. Hird and L. Zack, Nilpotent Lie and Leibniz algebras, Comm. Algebra, 42(6) (2014), 2404-2410.
  • L. Bosko-Dunbar, J. D. Dunbar, J. T. Hird and K. Stagg, Solvable Leibniz algebras with Heisenberg nilradical, Comm. Algebra, 43(6) (2015), 2272-2281.
  • L. Bosko, A. Hedges, J. T. Hird, N. Schwartz and K. Stagg, Jacobson's refinement of Engel's theorem for Leibniz algebras, Involve, 4(3) (2011), 293-296.
  • K. Bugg, A. Hedges, M. Lee, B. Morell, D. Scofield and S. McKay Sullivan, Cyclic Leibniz algebras, To appear, arXiv:1402.5821 [math.RA], (2014).
  • I. Demir, Classification of 5-dimensional complex nilpotent Leibniz algebras, Representations of Lie algebras, quantum groups and related topics, Contemp. Math., Amer. Math. Soc., Providence, RI, 713 (2018), 95-119.
  • I. Demir, K. C. Misra and E. Stitzinger, On some structures of Leibniz algebras, Recent advances in representation theory, quantum groups, algebraic geometry, and related topics, Contemp. Math., Amer. Math. Soc., Providence, RI, 623 (2014), 41-54.
  • I. Demir, K. C. Misra and E. Stitzinger, On classi cation of four-dimensional nilpotent Leibniz algebras, Comm. Algebra, 45(3) (2017), 1012-1018.
  • I. A. Karimjanov, A. Kh. Khudoyberdiyev and B. A. Omirov, Solvable Leibniz algebras with triangular nilradicals, Linear Algebra Appl., 466 (2015), 530-546.
  • A. Kh. Khudoyberdiyev, I. S. Rakhimov and Sh. K. Said Husain, On classification of 5-dimensional solvable Leibniz algebras, Linear Algebra Appl., 457 (2014), 428-454.
  • J.-L. Loday, Une version non commutative des algebres de Lie: les algebres de Leibniz, Enseign. Math., 39 (1993), 269-293.
  • E. L. Stitzinger, Minimal Nonnilpotent Solvable Lie Algebras, Proc. Amer. Math. Soc., 28(1) (1971), 47-49.
  • D. Towers, Lie algebras all of whose proper subalgebras are nilpotent, Linear Algebra Appl., 32 (1980), 61-73.

MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS

Year 2020, , 187 - 192, 14.07.2020
https://doi.org/10.24330/ieja.768254

Abstract

We classify all nonnilpotent, solvable Leibniz algebras with the property that all proper subalgebras are nilpotent. This generalizes the work of [E. L. Stitzinger, Proc. Amer. Math. Soc., 28(1)(1971), 47-49] and [D. Towers, Linear Algebra Appl., 32(1980), 61-73] in Lie algebras. We show several examples which illustrate the differences between the Lie and Leibniz results. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

References

  • D. W. Barnes, Some theorems on Leibniz algebras, Comm. Algebra, 39(7) (2011), 2463-2472.
  • C. Batten-Ray, A. Combs, N. Gin, A. Hedges, J. T. Hird and L. Zack, Nilpotent Lie and Leibniz algebras, Comm. Algebra, 42(6) (2014), 2404-2410.
  • L. Bosko-Dunbar, J. D. Dunbar, J. T. Hird and K. Stagg, Solvable Leibniz algebras with Heisenberg nilradical, Comm. Algebra, 43(6) (2015), 2272-2281.
  • L. Bosko, A. Hedges, J. T. Hird, N. Schwartz and K. Stagg, Jacobson's refinement of Engel's theorem for Leibniz algebras, Involve, 4(3) (2011), 293-296.
  • K. Bugg, A. Hedges, M. Lee, B. Morell, D. Scofield and S. McKay Sullivan, Cyclic Leibniz algebras, To appear, arXiv:1402.5821 [math.RA], (2014).
  • I. Demir, Classification of 5-dimensional complex nilpotent Leibniz algebras, Representations of Lie algebras, quantum groups and related topics, Contemp. Math., Amer. Math. Soc., Providence, RI, 713 (2018), 95-119.
  • I. Demir, K. C. Misra and E. Stitzinger, On some structures of Leibniz algebras, Recent advances in representation theory, quantum groups, algebraic geometry, and related topics, Contemp. Math., Amer. Math. Soc., Providence, RI, 623 (2014), 41-54.
  • I. Demir, K. C. Misra and E. Stitzinger, On classi cation of four-dimensional nilpotent Leibniz algebras, Comm. Algebra, 45(3) (2017), 1012-1018.
  • I. A. Karimjanov, A. Kh. Khudoyberdiyev and B. A. Omirov, Solvable Leibniz algebras with triangular nilradicals, Linear Algebra Appl., 466 (2015), 530-546.
  • A. Kh. Khudoyberdiyev, I. S. Rakhimov and Sh. K. Said Husain, On classification of 5-dimensional solvable Leibniz algebras, Linear Algebra Appl., 457 (2014), 428-454.
  • J.-L. Loday, Une version non commutative des algebres de Lie: les algebres de Leibniz, Enseign. Math., 39 (1993), 269-293.
  • E. L. Stitzinger, Minimal Nonnilpotent Solvable Lie Algebras, Proc. Amer. Math. Soc., 28(1) (1971), 47-49.
  • D. Towers, Lie algebras all of whose proper subalgebras are nilpotent, Linear Algebra Appl., 32 (1980), 61-73.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Lindsey Bosko-dunbar This is me

Jonathan D. Dunbar This is me

J. T. Hırd This is me

Kristen Stagg Rovıra This is me

Publication Date July 14, 2020
Published in Issue Year 2020

Cite

APA Bosko-dunbar, L., Dunbar, J. D., Hırd, J. T., Rovıra, K. S. (2020). MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS. International Electronic Journal of Algebra, 28(28), 187-192. https://doi.org/10.24330/ieja.768254
AMA Bosko-dunbar L, Dunbar JD, Hırd JT, Rovıra KS. MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS. IEJA. July 2020;28(28):187-192. doi:10.24330/ieja.768254
Chicago Bosko-dunbar, Lindsey, Jonathan D. Dunbar, J. T. Hırd, and Kristen Stagg Rovıra. “MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS”. International Electronic Journal of Algebra 28, no. 28 (July 2020): 187-92. https://doi.org/10.24330/ieja.768254.
EndNote Bosko-dunbar L, Dunbar JD, Hırd JT, Rovıra KS (July 1, 2020) MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS. International Electronic Journal of Algebra 28 28 187–192.
IEEE L. Bosko-dunbar, J. D. Dunbar, J. T. Hırd, and K. S. Rovıra, “MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS”, IEJA, vol. 28, no. 28, pp. 187–192, 2020, doi: 10.24330/ieja.768254.
ISNAD Bosko-dunbar, Lindsey et al. “MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS”. International Electronic Journal of Algebra 28/28 (July 2020), 187-192. https://doi.org/10.24330/ieja.768254.
JAMA Bosko-dunbar L, Dunbar JD, Hırd JT, Rovıra KS. MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS. IEJA. 2020;28:187–192.
MLA Bosko-dunbar, Lindsey et al. “MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS”. International Electronic Journal of Algebra, vol. 28, no. 28, 2020, pp. 187-92, doi:10.24330/ieja.768254.
Vancouver Bosko-dunbar L, Dunbar JD, Hırd JT, Rovıra KS. MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS. IEJA. 2020;28(28):187-92.