Simple finite-dimensional right-alternative unital superalgebras

S. V. Pchelintsev ; O. V. Shashkov

Izvestiya. Mathematics , Tome 82 (2018) no. 3, pp. 578-595.

Résumé

We study simple finite-dimensional right-alternative unital superalgebras over a field of characteristic zero. We prove that if the even part of a superalgebra is associative and commutative, then the superalgebra is of Abelian type. The classification of such superalgebras is known.

Détail

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

Bibliographie

[1] 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

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

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

[4] 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

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

[6] 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

[7] S. V. Pchelintsev, O. V. Shashkov, “Simple finite-dimensional right-alternative superalgebras with semisimple strongly associative even part”, Sb. Math., 208:2 (2017), 223–236 | DOI | DOI | MR | Zbl

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

[9] J. Picanço da Silva, L. S. I. Murakami, I. Shestakov, “On right alternative syperalgebras”, Comm. Algebra, 44:1 (2016), 240–252 | DOI | MR | Zbl

[10] S. V. Pchelintsev, O. V. Shashkov, “Simple finite-dimensional right alternative superalgebras with unitary even part over a field of characteristic $0$”, Math. Notes, 100:4 (2016), 589–596 | DOI | DOI | MR | Zbl

[11] S. V. Pchelintsev, O. V. Shashkov, “Simple right alternative superalgebras of Abelian type whose even part is a field”, Izv. Math., 80:6 (2016), 1231–1241 | DOI | DOI | MR | Zbl

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

[13] 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

[14] 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

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

[16] 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

[17] N. Jacobson, The theory of rings, Amer. Math. Soc. Math. Surveys, 2, Amer. Math. Soc., New York, 1943, vi+150 pp. | MR | Zbl

[18] T. Anderson, “Hereditary radicals and derivations of algebras”, Canad. J. Math., 21 (1969), 372–377 | DOI | MR | Zbl

[19] A. A. Albert, “On right alternative algebras”, Ann. of Math. (2), 50:2 (1949), 318–328 | DOI | MR | Zbl