Geodesic
Simple finite-dimensional right-alternative superalgebras of Abelian type of characteristic zero
Izvestiya. Mathematics , Tome 79 (2015) no. 3, pp. 554-580.

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We classify simple finite-dimensional right-alternative superalgebras $A=A_0\oplus A_1$ over a field of characteristic zero in which the even part $A_0$ is associative and commutative, while $A_1$ is an associative $A_0$-bimodule. We prove that every such superalgebra $A=A_0\oplus A_1$ is obtained by doubling the semisimple even part $A_0$, and the multiplication in $A$ is defined using a suitable automorphism and a linear operator acting on $A_0$.
Keywords: simple superalgebra, right-alternative superalgebra. 
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S. V. Pchelintsev; O. V. Shashkov. Simple finite-dimensional right-alternative superalgebras of Abelian type of characteristic zero. Izvestiya. Mathematics , Tome 79 (2015) no. 3, pp. 554-580. http://geodesic.mathdoc.fr/item/IM2_2015_79_3_a4/

[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., 213 (1964), 187–199 | DOI | MR | Zbl

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

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

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

[6] M. L. Racine, E. I. Zelmanov, “Simple Jordan superalgebras”, Non-associative algebra and its applications (Oviedo, 1993), Math. Appl., 303, Kluwer Acad. Publ., Dordrecht, 1994, 344–349 | MR | Zbl

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

[8] V. N. Zhelyabin, I. P. Shestakov, “Simple special Jordan superalgebras with associative even part”, Siberian Math. J., 45:5 (2004), 860–882 | DOI | MR | Zbl

[9] A. P. Pozhidaev, I. P. Shestakov, “Noncommutative Jordan superalgebras of degree $n>2$”, Algebra and Logic, 49:1 (2010), 18–42 | DOI | MR | Zbl

[10] 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

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

[12] J. P. Silva, L. S. I. Murakami, I. P. Shestakov, “On right alternative superalgebras”, Comm. Algebra (to appear)

[13] K. McCrimmon, “Speciality and non-speciality of two Jordan superalgebras”, J. Algebra, 149:2 (1992), 326–351 | DOI | MR | Zbl

[14] S. V. Pchelintsev, I. P. Shestakov, “Prime $(-1,1)$ and Jordan monsters and superalgebras of vector type”, J. Algebra, 423 (2015), 54–86 ; preprint MPIM 13-55, Preprint Series, Max-Planck-Institut für Mathematik, Bonn, 2013, 34 pp. http://www.mpim-bonn.mpg.de/node/263 | DOI | MR | Zbl

[15] 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., New York–London, 1982, xi+371 pp. | MR | MR | Zbl | Zbl

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

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

[18] I. M. Miheev, “Simple right alternative rings”, Algebra and Logic, 16:6 (1977), 456–476 | DOI | MR | Zbl

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

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