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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

New examples of simple Jordan superalgebras over an arbitrary field of characteristic 0


Author: V. N. Zhelyabin
Translated by: N. B. Lebedinskaya
Original publication: Algebra i Analiz, tom 24 (2012), nomer 4.
Journal: St. Petersburg Math. J. 24 (2013), 591-600
MSC (2010): Primary 16W10; Secondary 17A15
Published electronically: May 24, 2013
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Abstract: In a joint paper with the author, I. P. Shestakov constructed a new example of a unital simple special Jordan superalgebra over the real number field. It turned out that this superalgebra is a subsuperalgebra of a Jordan superalgebra of the vector type $ J(\Gamma ,D)$, but it is not isomorphic to a superalgebra of this type. Moreover, the superalgebra of quotients of the constructed superalgebra is isomorphic to a Jordan superalgebra of vector type. Later, a similar example was constructed for Jordan superalgebras over a field of characteristic 0 in which the equation $ t^2+1=0$ is unsolvable. In the present paper, an example is given for a Jordan superalgebra with the same properties over an arbitrary field of characteristic 0. A similar example was discovered also for a Cheng-Kac superalgebra.


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Additional Information

V. N. Zhelyabin
Affiliation: S. L. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences 4, Academician Koptyug prospect, Novosibirsk 630090, Russia; Novosibirsk State University, Pirogov street 2, Novosibirsk, 630090, Russia
Email: vicnic@math.nsc.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01255-6
PII: S 1061-0022(2013)01255-6
Keywords: Jordan superalgebra, $(-1,1)$-superalgebra, superalgebra of vector type, differentiably simple algebra, polynomial algebra, projective module
Received by editor(s): September 13, 2010
Published electronically: May 24, 2013
Additional Notes: Supported by the RFBR grant 09-01-00157, by the analytic Departmental Special Program “Development of the scientific potential of Higher School” of the Federal Educational Agency (project 2.1.1.419), by the Special Federal Program “Scientific and Pedagogical Staff of Innovative Russia for 2009–2013” (state contracts nos. 02.740.11.0429, 02.740.11.5191, 14.740.11.0346)
Article copyright: © Copyright 2013 American Mathematical Society