Research Article
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Year 2020, Volume 28, Issue 28, 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, Volume 28, Issue 28, 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.

Details

Primary Language English
Subjects Mathematics
Journal Section Articles
Authors

Lindsey BOSKO-DUNBAR This is me (Primary Author)
St. Norbert College
United States


Jonathan D. DUNBAR This is me
St. Norbert College
United States


J. T. HIRD This is me
West Virginia University, Institute of Technology
United States


Kristen Stagg ROVIRA This is me
California State University, Dominguez Hills
United States

Publication Date July 14, 2020
Published in Issue Year 2020, Volume 28, Issue 28

Cite

Bibtex @research article { ieja768254, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2020}, volume = {28}, number = {28}, pages = {187 - 192}, doi = {10.24330/ieja.768254}, title = {MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS}, key = {cite}, author = {Bosko-dunbar, Lindsey and Dunbar, Jonathan D. and Hırd, J. T. and Rovıra, Kristen Stagg} }
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 . DOI: 10.24330/ieja.768254
MLA Bosko-dunbar, L. , Dunbar, J. D. , Hırd, J. T. , Rovıra, K. S. "MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS" . International Electronic Journal of Algebra 28 (2020 ): 187-192 <https://dergipark.org.tr/en/pub/ieja/issue/55997/768254>
Chicago Bosko-dunbar, L. , Dunbar, J. D. , Hırd, J. T. , Rovıra, K. S. "MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS". International Electronic Journal of Algebra 28 (2020 ): 187-192
RIS TY - JOUR T1 - MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS AU - LindseyBosko-dunbar, Jonathan D.Dunbar, J. T.Hırd, Kristen StaggRovıra Y1 - 2020 PY - 2020 N1 - doi: 10.24330/ieja.768254 DO - 10.24330/ieja.768254 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 187 EP - 192 VL - 28 IS - 28 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.768254 UR - https://doi.org/10.24330/ieja.768254 Y2 - 2022 ER -
EndNote %0 International Electronic Journal of Algebra MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS %A Lindsey Bosko-dunbar , Jonathan D. Dunbar , J. T. Hırd , Kristen Stagg Rovıra %T MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS %D 2020 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 28 %N 28 %R doi: 10.24330/ieja.768254 %U 10.24330/ieja.768254
ISNAD Bosko-dunbar, Lindsey , Dunbar, Jonathan D. , Hırd, J. T. , Rovıra, Kristen Stagg . "MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS". International Electronic Journal of Algebra 28 / 28 (July 2020): 187-192 . https://doi.org/10.24330/ieja.768254
AMA Bosko-dunbar L. , Dunbar J. D. , Hırd J. T. , Rovıra K. S. MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS. IEJA. 2020; 28(28): 187-192.
Vancouver Bosko-dunbar L. , Dunbar J. D. , Hırd J. T. , Rovıra K. S. MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS. International Electronic Journal of Algebra. 2020; 28(28): 187-192.
IEEE L. Bosko-dunbar , J. D. Dunbar , J. T. Hırd and K. S. Rovıra , "MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS", International Electronic Journal of Algebra, vol. 28, no. 28, pp. 187-192, Jul. 2020, doi:10.24330/ieja.768254