Noncommutative Jordan superalgebras of degree $n>2$

A. P. Pozhidaev ; I. P. Shestakov

Algebra i logika, Tome 49 (2010) no. 1, pp. 26-59.

Résumé

We prove a coordinatization theorem for noncommutative Jordan superalgebras of degree $n>2$, describing such algebras. It is shown that the symmetrized Jordan superalgebra for a simple finite-dimensional noncommutative Jordan superalgebra of characteristic 0 and degree $n>1$ is simple. Modulo a “nodal” case, we classify central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0.

Détail

Keywords: noncommutative Jordan superalgebra, coordinatization theorem.

Bibliographie

[1] McCrimmon K., “Structure and representations of noncommutative Jordan algebras”, Trans. Am. Math. Soc., 121 (1966), 187–199 | MR | Zbl

[2] Jacobson N., “A theorem on the structure of Jordan algebras”, Proc. Natl. Acad. Sci. USA, 42 (1956), 140–147 | DOI | MR | Zbl

[3] Jacobson N., “A coordinatization theorem for Jordan algebras”, Proc. Natl. Acad. Sci. USA, 48 (1962), 1154–1160 | DOI | MR | Zbl

[4] Filippov V. T., Kharchenko V. K., Shestakov I. P., Nereshennye problemy teorii kolets i modulei, Dnestrovskaya tetrad, Izd-vo IM SO RAN, Novosibirsk, 1993 | MR | Zbl

[5] Sabinin L. V., Sbitneva L., Shestakov I. (eds.), Non-associative algebra and its applications, Proc. 5th int. conf. (Oaxtep, Mexico, July 27 – August 2, 2003), Lect. Notes Pure Appl. Math., 246, 2006 | MR | Zbl

[6] Schafer R. D., “Noncommutative Jordan algebras of characteristic 0”, Proc. Am. Math. Soc., 6 (1955), 472–475 | MR | Zbl

[7] Oehmke R. H., “On flexible algebras”, Ann. Math. (2), 68 (1958), 221–230 | DOI | MR | Zbl

[8] McCrimmon K., “Noncommutative Jordan rings”, Trans. Am. Math. Soc., 158 (1971), 1–33 | MR | Zbl

[9] Smith K. C., “Noncommutative Jordan algebras of capacity two”, Trans. Am. Math. Soc., 158:1 (1971), 151–159 | MR | Zbl

[10] Kokoris L. A., “Nodal non-commutative Jordan algebras”, Can. J. Math., 12 (1960), 488–492 | MR | Zbl

[11] Kokoris L. A., “Simple nodal noncommutative Jordan algebras”, Proc. Am. Math. Soc., 9 (1958), 652–654 | MR | Zbl

[12] Skosyrskii V. G., “Strogo pervichnye nekommutativnye iordanovy algebry”, Issledovaniya po teorii kolets i algebr, Tr. In-ta matem. SO AN SSSR, 16, Nauka, Novosibirsk, 1989, 131–163 | MR

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

[14] Kantor I.L., “Iordanovy i lievy superalgebry, opredelennye algebroi Puassona”, Algebra i Analiz, izd-vo TGU, Tomsk, 1989, 55–80 | MR

[15] Kaplansky I., “Superalgebras”, Pac. J. Math., 86 (1980), 93–98 | MR | Zbl

[16] Racine M., Zelmanov E., Classification of simple Jordan superalgebras with semisimple even part, preprint, 1998 | MR | Zbl

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

[18] Albert A. A., “Power-associative rings”, Trans. Am. Math. Soc., 64 (1948), 552–593 | MR | Zbl

[19] Albert A. A., “A theory of trace-admissible algebras”, Proc. Natl. Acad. Sci. USA, 35 (1949), 317–322 | DOI | MR | Zbl

[20] Cheng S.-J., “Differentiably simple Lie superalgebras and representations of semisimple Lie superalgebras”, J. Algebra, 173:1 (1995), 1–43 | DOI | MR | Zbl

[21] McCrimmon K., “The splittest Kac superalgebra $K_{10}$”, J. Algebra, 313:2 (2007), 554–589 | DOI | MR | Zbl

[22] Sib. Adv. Math., 9:2 (1999), 83–99 | MR | MR | Zbl | Zbl