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Year 2024, Volume: 36 Issue: 36, 89 - 100, 12.07.2024
https://doi.org/10.24330/ieja.1446322

Abstract

References

  • L. Auslander and J. Brezin, Almost algebraic Lie algebras, J. Algebra, 8 (1968), 295-313.
  • Sh. A. Ayupov and B. A. Omirov, On Leibniz algebras, Algebra and operator theory, Tashkent (1997), Kluwer Academic Publishers, (1998), 1-12.
  • Sh. A. Ayupov, B. Omirov and I. Rakhimov, Leibniz Algebras-Structure and Classification, CRC Press, Boca Raton, 2020.
  • D. W. Barnes, Some theorems on Leibniz algebras, Comm. Algebra, 39(7) (2011), 2463-2472.
  • C. Batten, L. Bosko-Dunbar, A. Hedges, J. T. Hird, K. Stagg and E. Stitzinger, A Frattini theory for Leibniz algebras, Comm. Algebra, 41(4) (2013), 1547-1557.
  • J. Feldvoss, Leibniz algebras as non-associative algebras, Nonassociative mathematics and its applications, Contemp. Math., 721 (2019), 115-149.
  • M. Jibladze and T. Pirashvili, Lie theory for symmetric Leibniz algebras, J. Homotopy Relat. Struct., 15(1) (2020), 167-183.
  • S. Siciliano and D. A. Towers, On the subalgebra lattice of a Leibniz algebra, Comm. Algebra, 50(1) (2022), 255-267.
  • E. L. Stitzinger, Frattini subalgebras of a class of solvable Lie algebras, Pacific J. Math., 34 (1970), 177-182.
  • D. A. Towers, A Frattini theory for algebras, Proc. London Math. Soc. (3), 27 (1973), 440-462.
  • D. A. Towers, Solvable Lie $A$-algebras, J. Algebra, 340 (2011), 1-12.
  • D. A. Towers, Leibniz $A$-algebras, Commun. Math., 28(2) (2020), 103-121.
  • D. A. Towers, On the nilradical of a Leibniz algebra, Comm. Algebra, 49(10) (2021), 4345-4347.
  • D. A. Towers and V. R. Varea, Further results on elementary Lie algebras and Lie $A$-algebras, Comm. Algebra, 41(4) (2013), 1432-1441.
  • N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, Interscience, New York-London, 1962.

Almost-reductive and almost-algebraic Leibniz algebra

Year 2024, Volume: 36 Issue: 36, 89 - 100, 12.07.2024
https://doi.org/10.24330/ieja.1446322

Abstract

This paper examines whether the concept of an almost-algebraic Lie algebra developed by Auslander and Brezin in
[J. Algebra, 8(1968), 295-313] can be introduced for Leibniz
algebras. Two possible analogues are considered: almost-reductive
and almost-algebraic Leibniz algebras. For Lie algebras these two
concepts are the same, but that is not the case for Leibniz
algebras, the class of almost-algebraic Leibniz algebras strictly
containing that of the almost-reductive ones. Various properties
of these two classes of algebras are obtained, together with some
relationships between $\phi$-free, elementary, $E$-algebras and
$A$-algebras.

References

  • L. Auslander and J. Brezin, Almost algebraic Lie algebras, J. Algebra, 8 (1968), 295-313.
  • Sh. A. Ayupov and B. A. Omirov, On Leibniz algebras, Algebra and operator theory, Tashkent (1997), Kluwer Academic Publishers, (1998), 1-12.
  • Sh. A. Ayupov, B. Omirov and I. Rakhimov, Leibniz Algebras-Structure and Classification, CRC Press, Boca Raton, 2020.
  • D. W. Barnes, Some theorems on Leibniz algebras, Comm. Algebra, 39(7) (2011), 2463-2472.
  • C. Batten, L. Bosko-Dunbar, A. Hedges, J. T. Hird, K. Stagg and E. Stitzinger, A Frattini theory for Leibniz algebras, Comm. Algebra, 41(4) (2013), 1547-1557.
  • J. Feldvoss, Leibniz algebras as non-associative algebras, Nonassociative mathematics and its applications, Contemp. Math., 721 (2019), 115-149.
  • M. Jibladze and T. Pirashvili, Lie theory for symmetric Leibniz algebras, J. Homotopy Relat. Struct., 15(1) (2020), 167-183.
  • S. Siciliano and D. A. Towers, On the subalgebra lattice of a Leibniz algebra, Comm. Algebra, 50(1) (2022), 255-267.
  • E. L. Stitzinger, Frattini subalgebras of a class of solvable Lie algebras, Pacific J. Math., 34 (1970), 177-182.
  • D. A. Towers, A Frattini theory for algebras, Proc. London Math. Soc. (3), 27 (1973), 440-462.
  • D. A. Towers, Solvable Lie $A$-algebras, J. Algebra, 340 (2011), 1-12.
  • D. A. Towers, Leibniz $A$-algebras, Commun. Math., 28(2) (2020), 103-121.
  • D. A. Towers, On the nilradical of a Leibniz algebra, Comm. Algebra, 49(10) (2021), 4345-4347.
  • D. A. Towers and V. R. Varea, Further results on elementary Lie algebras and Lie $A$-algebras, Comm. Algebra, 41(4) (2013), 1432-1441.
  • N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, Interscience, New York-London, 1962.
There are 15 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Articles
Authors

David A. Towers This is me

Early Pub Date March 3, 2024
Publication Date July 12, 2024
Published in Issue Year 2024 Volume: 36 Issue: 36

Cite

APA Towers, D. A. (2024). Almost-reductive and almost-algebraic Leibniz algebra. International Electronic Journal of Algebra, 36(36), 89-100. https://doi.org/10.24330/ieja.1446322
AMA Towers DA. Almost-reductive and almost-algebraic Leibniz algebra. IEJA. July 2024;36(36):89-100. doi:10.24330/ieja.1446322
Chicago Towers, David A. “Almost-Reductive and Almost-Algebraic Leibniz Algebra”. International Electronic Journal of Algebra 36, no. 36 (July 2024): 89-100. https://doi.org/10.24330/ieja.1446322.
EndNote Towers DA (July 1, 2024) Almost-reductive and almost-algebraic Leibniz algebra. International Electronic Journal of Algebra 36 36 89–100.
IEEE D. A. Towers, “Almost-reductive and almost-algebraic Leibniz algebra”, IEJA, vol. 36, no. 36, pp. 89–100, 2024, doi: 10.24330/ieja.1446322.
ISNAD Towers, David A. “Almost-Reductive and Almost-Algebraic Leibniz Algebra”. International Electronic Journal of Algebra 36/36 (July 2024), 89-100. https://doi.org/10.24330/ieja.1446322.
JAMA Towers DA. Almost-reductive and almost-algebraic Leibniz algebra. IEJA. 2024;36:89–100.
MLA Towers, David A. “Almost-Reductive and Almost-Algebraic Leibniz Algebra”. International Electronic Journal of Algebra, vol. 36, no. 36, 2024, pp. 89-100, doi:10.24330/ieja.1446322.
Vancouver Towers DA. Almost-reductive and almost-algebraic Leibniz algebra. IEJA. 2024;36(36):89-100.