Almost-reductive and almost-algebraic Leibniz algebra

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.

Keywords

• Leibniz algebra
• symmetric Leibniz algebra
• Frattini ideal
• $\phi$-free
• elementary
• $E$-algebra
• $A$-algebra
• almost-algebraic
• almost-reductive

References

1. L. Auslander and J. Brezin, Almost algebraic Lie algebras, J. Algebra, 8 (1968), 295-313.
2. Sh. A. Ayupov and B. A. Omirov, On Leibniz algebras, Algebra and operator theory, Tashkent (1997), Kluwer Academic Publishers, (1998), 1-12.
3. Sh. A. Ayupov, B. Omirov and I. Rakhimov, Leibniz Algebras-Structure and Classification, CRC Press, Boca Raton, 2020.
4. D. W. Barnes, Some theorems on Leibniz algebras, Comm. Algebra, 39(7) (2011), 2463-2472.
5. 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.
6. J. Feldvoss, Leibniz algebras as non-associative algebras, Nonassociative mathematics and its applications, Contemp. Math., 721 (2019), 115-149.
7. M. Jibladze and T. Pirashvili, Lie theory for symmetric Leibniz algebras, J. Homotopy Relat. Struct., 15(1) (2020), 167-183.
8. S. Siciliano and D. A. Towers, On the subalgebra lattice of a Leibniz algebra, Comm. Algebra, 50(1) (2022), 255-267.

9. E. L. Stitzinger, Frattini subalgebras of a class of solvable Lie algebras, Pacific J. Math., 34 (1970), 177-182.
10. D. A. Towers, A Frattini theory for algebras, Proc. London Math. Soc. (3), 27 (1973), 440-462.
11. D. A. Towers, Solvable Lie $A$-algebras, J. Algebra, 340 (2011), 1-12.
12. D. A. Towers, Leibniz $A$-algebras, Commun. Math., 28(2) (2020), 103-121.
13. D. A. Towers, On the nilradical of a Leibniz algebra, Comm. Algebra, 49(10) (2021), 4345-4347.
14. D. A. Towers and V. R. Varea, Further results on elementary Lie algebras and Lie $A$-algebras, Comm. Algebra, 41(4) (2013), 1432-1441.
15. N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, Interscience, New York-London, 1962.