Simple finite-dimensional right-alternative superalgebras with semisimple strongly associative even part

S. V. Pchelintsev   ; O. V. Shashkov

Sbornik. Mathematics, Tome 208 (2017) no. 2, pp. 223-236

Résumé

It is shown that a simple finite-dimensional right-alternative unital superalgebra with semisimple strongly associative even part over a field of characteristic $\ne 2$ is either non-associative, or a superalgebra of Abelian type. A classification of such superalgebras over an algebraically closed field is given. Bibliography: 21 titles.

Détail

Keywords: simple superalgebra, right-alternative superalgebra, superalgebra of Abelian type.

Bibliographie

[1] E. I. Zel'manov, I. P. Shestakov, “Prime alternative superalgebras and nilpotence of the radical of a free alternative algebra”, Math. USSR-Izv., 37:1 (1991), 19–36 | DOI | MR | Zbl

[2] C. T. C. Wall, “Graded Brauer groups”, J. Reine Angew. Math., 1964:213 (1964), 187–199 | DOI | MR | Zbl

[3] V. G. Kac, “Classification of simple $Z$-graded Lie superalgebras and simple Jordan superalgebras”, Comm. Algebra, 5:13 (1977), 1375–1400 | DOI | MR | Zbl

[4] I. L. Kantor, “Jordan and Lie superalgebras determined by a Poisson algebra”, Algebra and analysis (Tomsk, 1989), Amer. Math. Soc. Transl. Ser. 2, 151, Amer. Math. Soc., Providence, RI, 1992, 55–80 | DOI | MR | Zbl

[5] I. P. Shestakov, “Prime alternative superalgebras of arbitrary characteristic”, Algebra and Logic, 36:6 (1997), 389–412 | DOI | MR | Zbl

[6] I. P. Shestakov, “Simple $(-1,1)$-superalgebras”, Algebra and Logic, 37:6 (1998), 411–422 | DOI | MR | Zbl

[7] M. L. Racine, E. I. Zelmanov, “Simple Jordan superalgebras with semisimple even part”, J. Algebra, 270:2 (2003), 374–444 | DOI | MR | Zbl

[8] C. Martinez, E. Zelmanov, “Simple finite-dimensional Jordan superalgebras of prime characteristic”, J. Algebra, 236:2 (2001), 575–629 | DOI | MR | Zbl

[9] A. P. Pozhidaev, I. P. Shestakov, “Simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0”, Siberian Math. J., 54:2 (2013), 301–316 | DOI | MR | Zbl

[10] V. N. Zhelyabin, “Simple special Jordan superalgebras with associative nil-semisimple even part”, Algebra and Logic, 41:3 (2002), 152–172 | DOI | MR | Zbl

[11] V. N. Zhelyabin, “Simple Jordan superalgebras with associative nil-semisimple even part”, Siberian Math. J., 57:6 (2016), 987–1001 | DOI

[12] “Dniester notebook: unsolved problems in the theory of rings and modules”, Non-associative algebra and its applications, Lect. Notes Pure Appl. Math., 246, eds. V. T. Filippov, I. P. Shestakov, V. K. Kharchenko, Chapman Hall/CRC, Boca Raton, FL, 2006, 461–516 | DOI | MR | MR | Zbl | Zbl

[13] A. A. Albert, “The structure of right alternative algebras”, Ann. of Math. (2), 59:3 (1954), 408–417 | DOI | MR | Zbl

[14] V. G. Skosyrskii, “Right alternative algebras”, Algebra and Logic, 23:2 (1984), 131–136 | DOI | MR | Zbl

[15] S. V. Pchelintsev, I. P. Shestakov, “Prime $(-1,1)$ and Jordan monsters and superalgebras of vector type”, J. Algebra, 423 (2015), 54–86 | DOI | MR | Zbl

[16] S. V. Pchelintsev, O. V. Shashkov, “Simple finite-dimensional right-alternative superalgebras of Abelian type of characteristic zero”, Izv. Math., 79:3 (2015), 554–580 | DOI | DOI | MR | Zbl

[17] S. V. Pchelintsev, O. V. Shashkov, “Simple right alternative superalgebras of Abelian type whose even parts are fields”, Izv. Math., 80:6 (2016), in press | DOI | DOI

[18] N. Jacobson, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publ., 39, Amer. Math. Soc., Providence, R.I., 1968, x+453 pp. | MR | Zbl

[19] K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov, A. I. Shirshov, Rings that are nearly associative, Pure Appl. Math., 104, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York–London, 1982, xi+371 pp. | MR | MR | Zbl | Zbl

[20] E. Kleinfeld, “Right alternative rings”, Proc. Amer. Math. Soc., 4:6 (1953), 939–944 | DOI | MR | Zbl

[21] S. Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965, xvii+508 pp. | MR | Zbl | Zbl