Finite-dimensional Leibniz algebra representations of $\mathfrak{sl}_2$

Abstract

All finite-dimensional Leibniz algebra bimodules of a Lie algebra $\mathfrak{sl}_2$ over a field of characteristic zero are described.

Keywords

• Leibniz algebra
• Leibniz algebra bimodule
• Finite-dimensional Leibniz algebra

References

1. [1] D. Barnes, Some Theorems on Leibniz algebras, Comm. Algebra, 39, 2463–2472, 2011.
2. [2] A. Bloh, On a generalization of the concept of Lie algebra, Dokl. Akad. Nauk SSSR, 165, 471–473, 1965.
3. [3] A.M. Bloh, A certain generalization of the concept of Lie algebra, Uch. Zap. Moskov. Gos. Ped. Inst. 375, 9–20, 1971 (in Russian).
4. [4] A.S. Dzhumadil’daev and A.S. Abdukassymova, Leibniz algebras in characteristic p, C.R.Acad.sci.Paris Ser. I Math. 332 (12), 1047–1052, 2001.
5. [5] P. Gabriel, Unzerlegbare Darstellungen, I, Manuscripta Math. 6, 71–103, 1972.
6. [6] N. Jacobson, Lie algebras, Interscience Publishers, Wiley, New York, 1962.
7. [7] T. Kurbanbaev and R. Turdibaev, Some Leibniz bimodules of sl2, J. Algebra Appl. 19 (4), 2050064, 2020.
8. [8] J.-L. Loday, Cyclic homology, Grundl. Math. Wiss. Bd. 301, Springer-Verlag, Berlin, 1992.

9. [9] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296, 139–158, 1993.
10. [10] J.-L. Loday and T. Pirashvili, Leibniz representations of Lie algebras, J. Algebra, 181 (2), 414–425, 1996.
11. [11] R. Martinez-Villa, Algebras Stably Equivalent to l-Hereditary, Springer Lecture Notes in Math., 832, pp. 396–431, Springer-Verlag, New York/Berlin, 1980.