Some constructions for Jordan superalgebras with associative even part

Zhelyabin, V.; Zakharov, A.

doi:10.1090/spmj/1446

Some constructions for Jordan superalgebras with associative even part
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by V. N. Zhelyabin and A. S. Zakharov
Translated by: N. B. Lebedinskaya

St. Petersburg Math. J. 28 (2017), 197-208

DOI: https://doi.org/10.1090/spmj/1446

Published electronically: February 15, 2017

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Abstract:

A construction is presented that enables one to build Jordan superalgebras by using Jordan superalgebras with associative even part. This is a generalization of a known construction of the addition of an odd variable in the case of superalgebras of Jordan brackets. Previously, this construction was described for simple special Jordan algebras with associative even part.

References

I. L. Kantor, Jordan and Lie superalgebras determined by a Poisson algebra, Algebra and analysis (Tomsk, 1989) Amer. Math. Soc. Transl. Ser. 2, vol. 151, Amer. Math. Soc., Providence, RI, 1992, pp. 55–80. MR 1191172, DOI 10.1090/trans2/151/03
I. P. Shestakov, Superalgebras and counterexamples, Sibirsk. Mat. Zh. 32 (1991), no. 6, 187–196, 207 (Russian); English transl., Siberian Math. J. 32 (1991), no. 6, 1052–1060 (1992). MR 1156760, DOI 10.1007/BF00971214
Daniel King and Kevin McCrimmon, The Kantor construction of Jordan superalgebras, Comm. Algebra 20 (1992), no. 1, 109–126. MR 1145328, DOI 10.1080/00927879208824339
Kevin McCrimmon, Speciality and nonspeciality of two Jordan superalgebras, J. Algebra 149 (1992), no. 2, 326–351. MR 1172433, DOI 10.1016/0021-8693(92)90020-M
Daniel King and Kevin McCrimmon, The Kantor doubling process revisited, Comm. Algebra 23 (1995), no. 1, 357–372. MR 1311793, DOI 10.1080/00927879508825225
Consuelo Martínez, Ivan Shestakov, and Efim Zelmanov, Jordan superalgebras defined by brackets, J. London Math. Soc. (2) 64 (2001), no. 2, 357–368. MR 1853456, DOI 10.1112/S0024610701002290
V. N. Zhelyabin, Simple special Jordan superalgebras with an associative nil-semisimple even part, Algebra Logika 41 (2002), no. 3, 276–310, 386–387 (Russian, with Russian summary); English transl., Algebra Logic 41 (2002), no. 3, 152–172. MR 1934537, DOI 10.1023/A:1016072808189
V. N. Zhelyabin and I. P. Shestakov, Simple special Jordan superalgebras with an associative even part, Sibirsk. Mat. Zh. 45 (2004), no. 5, 1046–1072 (Russian, with Russian summary); English transl., Siberian Math. J. 45 (2004), no. 5, 860–882. MR 2108503, DOI 10.1023/B:SIMJ.0000042476.85436.a3
V. N. Zhelyabin, Differential algebras and simple Jordan superalgebras, Mat. Tr. 12 (2009), no. 2, 41–51 (Russian, with Russian summary). MR 2599424
V. N. Zhelyabin, New examples of simple Jordan superalgebras over an arbitrary field of characteristic zero, Algebra i Analiz 24 (2012), no. 4, 84–96 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 24 (2013), no. 4, 591–600. MR 3088008, DOI 10.1090/S1061-0022-2013-01255-6
Nicoletta Cantarini and Victor G. Kac, Classification of linearly compact simple Jordan and generalized Poisson superalgebras, J. Algebra 313 (2007), no. 1, 100–124. MR 2326139, DOI 10.1016/j.jalgebra.2006.10.040
V. N. Zhelyabin, Examples of prime Jordan superalgebras of vector type and of superalgebras of Cheng-Kac type, Sibirsk. Mat. Zh. 54 (2013), no. 1, 49–56 (Russian, with Russian summary); English transl., Sib. Math. J. 54 (2013), no. 1, 33–39. MR 3089325, DOI 10.1134/S0037446613010059
Richard G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264–277. MR 143225, DOI 10.1090/S0002-9947-1962-0143225-6
V. N. Zhelyabin, Dual coalgebras of Jordan bialgebras and superalgebras, Sibirsk. Mat. Zh. 46 (2005), no. 6, 1302–1315 (Russian, with Russian summary); English transl., Siberian Math. J. 46 (2005), no. 6, 1050–1061. MR 2195030, DOI 10.1007/s11202-005-0099-6
Edward C. Posner, Differentiably simple rings, Proc. Amer. Math. Soc. 11 (1960), 337–343. MR 113908, DOI 10.1090/S0002-9939-1960-0113908-6