eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Induced 3-Lie algebras, superalgebras and induced representations; pp. 116–133

Viktor Abramov, Priit Lätt

We construct 3-Lie superalgebras on a commutative superalgebra by means of involution and even degree derivation. We construct a representation of induced 3-Lie algebras and superalgebras by means of a representation of initial (binary) Lie algebra, trace and supertrace. We show that the induced representation of 3-Lie algebra, that we constructed, is a representation by traceless matrices, that is, lies in the Lie algebra sl(V ), where V is a representation space. In the case of 2-dimensional representation we find conditions under which the induced representation of induced 3-Lie algebra is irreducible. We give the example of irreducible representation of induced 3-Lie algebra of 2nd order complex matrices.


1. Abramov, V. Super 3-Lie Algebras Induced by Super Lie Algebras. Adv. Appl. Clifford Algebras, 2017, 27, 9–16.

2. Abramov, V. Matrix 3-Lie superalgebras and BRST supersymmetry. Int. J. Geom. Methods Mod. Phys., 2017, 14(11), 1750160.

3. Abramov, V. 3-Lie Superalgebras Induced by Lie Superalgebras. Axioms, 2019, 8(1), 21.

4. Arnlind J., Makhlouf A., and Silvestrov, S. Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras. J. Math. Phys., 2010, 51, 043515.

5. Arnlind J., Makhlouf A., and Silvestrov, S. Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras. J. Math. Phys., 2011, 52, 123502.

6. Awata, H., Li, M., Minic, D., and Yaneya, T. On the quantization of Nambu brackets. JHEP02, 2001, 013.

7. Bai, C., Guo, L., and Sheng, Y. Bialgebras, the classical Yang-Baxter equation and Manin triples for 3-Lie algebras. Adv. Theor. Math. Phys., 2016, 23(1), 27–74.

8. Bai, R. and Wu, Y. Constructions of 3-Lie algebras. Linear Multilinear Algebra, 2015, 63(11), 2171–2186.

9. Filippov, V. T. n-Lie algebras. Siberian Math. J., 1985, 26, 879–891.

10. Guan, B., Chen, L., and Sun, B. 3-ary Hom-Lie Superalgebras Induced by Hom-Lie Superalgebras. Adv. Appl. Clifford Algebras, 2017, 27, 3063–3082.

11. Kitouni A. On (1)-Lie induced by n-Lie algebras. PhD Thesis University of Upper Alsace, Mulhouse, 2015.

12. Kitouni A, Makhlouf A., and Silvestrov, S. On (1)-Hom-Lie algebras induced by n-Hom-Lie algebras. Georgian Math. J., 2016, 23, 75–95.

13. Nambu, Y. Generalized Hamiltonian mechanics. Phys. Rev. D., 1973, 7, 2405–2412.



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