Abstract
We describe the simple special unital Jordan superalgebras with associative even part A whose odd part M is an associative module over A. We prove that each of these superalgebras, not isomorphic to a superalgebra of nondegenerate bilinear superform, is isomorphically embedded into a twisted Jordan superalgebra of vector type. We exhibit a new example of a simple special Jordan superalgebra. We also describe the superalgebras such that M∩ [A,M]≠ 0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kac V., “Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras,” Comm. Algebra, 5, 1375-1400 (1977).
Kantor I. L., “Jordan and Lie superalgebras defined by Poisson algebra,” in: Amer. Math. Soc. Transl. Ser. 2. Vol. 151, Amer. Math. Soc., Providence, RI, 1992, pp. 55-80.
Racine M. and Zelmanov E., “Simple Jordan superalgebras with semisimple even part,” J. Algebra, 270, No. 2, 374-444 (2003).
King D. and McCrimmon K., “The Kantor construction of Jordan superalgebras,” Comm. Algebra, 20, No. 1, 109-126 (1992).
Martinez C. and Zelmanov E., “Simple finite-dimensional Jordan superalgebras of prime characteristic,” J. Algebra, 236, 575-629 (2001).
Shestakov I. P. and Zelmanov E. I., “Prime alternative superalgebras and nilpotency of a radical of a free alternative algebra,” Izv. Akad. Nauk SSSR Ser. Mat., 54, No. 4, 676-693 (1990).
Shestakov I. P., “Prime alternative superalgebras of an arbitrary characteristic,” Algebra i Logika, 36, No. 6, 701-731 (1997).
Shestakov I. P., “Simple superalgebras of the kind (-1, 1),” Algebra i Logika, 37, No. 6, 721-739 (1998).
Zhelyabin V. N., “Simple special Jordan superalgebras with associative nilsemisimple even part,” Algebra i Logika, 41, No. 3, 276-310 (2002).
Gonzalez S., Lopez-Diaz M. C., Martinez C., and Shestakov I. P., “Bernstein superalgebras and superbimodules,” J. Algebra, 212, No. 1, 119-131 (1999).
King D. and McCrimmon K., “The Kantor doubling process revisited,” Comm. Algebra, 23, No. 1, 357-372 (1995).
Shuen Yuan, “Differentiable simple rings of prime characteristic,” Duke Math. J., 31, No. 4, 623-630 (1964).
Posner E. C., “Differentiable simple rings,” Proc. Amer. Math. Soc., 11, No. 3, 337-343 (1968).
Bourbaki N., Commutative Algebra [Russian translation], Nauka, Moscow (1971).
Suslin A. A., “On the structure of the special linear group over polynomial rings,” Izv. Akad. Nauk SSSR Ser. Mat., 41, No. 2, 235-252 (1977).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zhelyabin, V.N., Shestakov, I.P. Simple Special Jordan Superalgebras with Associative Even Part. Siberian Mathematical Journal 45, 860–882 (2004). https://doi.org/10.1023/B:SIMJ.0000042476.85436.a3
Issue Date:
DOI: https://doi.org/10.1023/B:SIMJ.0000042476.85436.a3