Abstract
The classification of Lie superalgebras of small dimensions is well known. Here we set out a similar task regarding Mal'tsev superalgebras. It is established that there exist a number of diverse types of non-Lie Mal'tsev superalgebras of dimension 4. One of the types of 3-dimensional non-Lie Mal'tsev superalgebras allows us to construct a number of interesting examples in the theory of ordinary (finite-dimensional) Mal'tsev algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Albuquerque, Contribucoes para a teoria das superalgebras de Malcev, Doctoral Thesis, University of Coimbra (1993).
H. Albuquerque, “Malcev superalgebras,”Proc. III International Conference on Non Associative Algebra and Its Applications, Oviedo, 1993, Kluwer (1994), pp. 1–7.
H. Albuquerque and A. Elduque, “Malcev superalgebras with trivial nucleus,”Comm. Alg. 21, No. 9, 3147–3164 (1993).
H. Albuquerque and A. Elduque, “Engel's theorem for Malcev superalgebras,”Comm. Alg.,22, No. 14, 5689–5701 (1994).
A. Elduque and I. P. Shestakov, “On Malcev superalgebras with trivial Lie nucleus,”Nova J. Alg. Geom.,2, No. 4, 389–391 (1993).
A. Elduque and I. P. Shestakov, “Irreducible non-Lie modules for Malcev superalgebras,”J. Alg.,173, No. 3, 622–637 (1995).
I. P. Shestakov and A. Elduque, “Prime non-Lie modules for Mal'tsev superalgebras,”Algebra Logika,33, No. 4, 448–465 (1994).
I. P. Shestakov, “Prime Mal'tsev superalgebras,”Mat. Sb.,182, No. 9, 1357–1366 (1991).
J. Patera, R. T. Sharp, and P. Winternitz, “Invariants of real low dimension Lie algebras,”J. Math. Phys.,17, No. 6, 986–994.
N. Backhouse, “A classification of four dimensional Lie superalgebras,”J. Math. Phys.,19, No. 11, 2400–2402.
E. N. Kuzmin, “Mal'tsev algebras of dimension five over a field of characteristic zero,”Algebra Logika, 9, No. 6, 691–700 (1970).
A. A. Sagle, “Malcev algebras,”Trans. Am. Math. Soc.,101, No. 3, 426–458 (1961).
A. Elduque, “On Malcev algebras in which all subideals are ideals,”Proc. Royal Soc. Edinburgh,101A, 353–363 (1985).
Yu. A. Medvedev and E. I. Zelmanov, “Some counterexamples in the theory of Jordan algebras,” inNonassociative Algebraic Models, (S. Gonzalez ed.), Nova Science, New York (1992), pp. 1–16.
I. P. Shestakov, “Superalgebras and counterexamples,”Sib. Mat. Zh.,32, No. 6, 187–196 (1991).
A. Elduque and A. A. ElMalek, “On the J-nucleus of a Malcev algebra,”Algebras Groups Geom.,3, 493–503 (1986).
E. N. Kuzmin, “The locally nilpotent radical of a Mal'tsev algebra satisfying thenth Engel condition,”Dokl. Akad. Nauk SSSR,177, No. 3, 508–510 (1967).
V. T. Filippov, “Zero divisors and nil elements in Mal'tsev algebras,”Algebra Logika,14, No. 2, 204–214 (1975).
V. T. Filippov, “On Engelian Mal'tsev algebras,”Algebra Logika,15, No. 1, 89–109 (1976).
V. T. Filippov, “Nilpotent ideals of Mal'tsev algebras,”Algebra Logika,18, No. 5, 599–613 (1979).
E. N. Kuzmin, “Mal'tsev algebras and their representations,”Algebra Logika,7, No. 4, 48–69 (1968).
E. N. Kuzmin, “Structure and representations of finite-dimensional Mal'tsev algebras,” inTheory of Rings and Algebras, Trudy Inst. Mat. SO AN SSSR, Vol. 16, Nauka, Novosibirsk (1989), pp. 75–101.
V. G. Kac, “Lie superalgebras,”Adv. Math.,26, No. 1, 8–96 (1977).
A. T. Gainov, “Binary Lie algebras of lower ranks”Algebra Logika, Seminar,2, No. 4, 21–40 (1963)
A. Elduque and H. C. Myung,Mutations of Alternative Algebras, Kluwer, Dordrecht (1994).
E. N. Kuzmin, “Binary Lie algebras of low dimensions,” to appear inAlgebra Logika.
Additional information
Supported by CMUC-JNICT.
Supported by DGA (PCB 6/91).
Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 629–654, November–December, 1996.
Rights and permissions
About this article
Cite this article
Albuquerque, H., Elduque, A. A classification of Mal'tsev superalgebras of small dimensions. Algebr Logic 35, 351–365 (1996). https://doi.org/10.1007/BF02366395
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02366395