Quasiregular radicals of nonassociative algebras

A. Yu. Golubkov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 24 (2023) no. 4, pp. 75-128.

Résumé

This paper considers constructions of the quasiregular radical of linear algebras based on its associative and Jordan definitions, and methods for constructing homogeneous radicals of Lie algebras with finite pregrading based on radicals of linear Jordan pairs that allow one to determine the Lie analogues of the quasiregular radical.

Bibliographie

[1] Beidar K. I., Pikhtilkov S. A., “O pervichnom radikale spetsialnykh algebr Li”, UMN, 49:1 (1994), 233 | MR | Zbl

[2] Golubkov A. Yu., “Lokalnaya konechnost algebr”, Fundament. i prikl. matem., 19:6 (2014), 25–75

[3] Golubkov A. Yu., “Konstruktsii spetsialnykh radikalov algebr”, Fundament. i prikl. matem., 20:1 (2015), 57–133

[4] Golubkov A. Yu., “Algebraicheskie algebry Li ogranichennoi stepeni”, Fundament. i prikl. matem., 22:5 (2019), 209–242

[5] Grishkov A. N., “O lokalnoi nilpotentnosti ideala algebry Li, porozhdennogo elementami 2-go poryadka”, Sib. matem. zhurn, 23:1 (1982), 181–182 | MR | Zbl

[6] Zhevlakov K. A., Slinko A. M., Shestakov I. P., Shirshov A. I., Koltsa, blizkie k assotsiativnym, Nauka, M., 1978 | MR

[7] Zelmanov E. I., “O pervichnykh iordanovykh algebrakh”, Algebra i logika, 18:2 (1979), 162–175 | MR | Zbl

[8] Zelmanov E. I., “Absolyutnye deliteli nulya v iordanovykh parakh i algebrakh Li”, Matem. sb., 112(154):4(8) (1980), 611–629 | MR | Zbl

[9] Zelmanov E. I., “Absolyutnye deliteli nulya i algebraicheskie iordanovy algebry”, Sib. matem. zhurn., 23:6 (1980), 100–116 | MR

[10] Zelmanov E. I., “Algebry Li s algebraicheskim prisoedinennym predstavleniem”, Matem. sb., 121(163):4(8) (1983), 545–561 | MR

[11] Zelmanov E. I., “Pervichnye iordanovy algebry II”, Sib. matem. zhurn., 24:1 (1983), 89–104 | MR | Zbl

[12] Zelmanov E. I., “Algebry Li s konechnoi graduirovkoi”, Matem. sb., 124(166):3(7) (1984), 353–392 | MR | Zbl

[13] Zelmanov E. I., “Kharakterizatsiya radikala Makkrimmona”, Sib. matem. zhurn., 25:5 (1984), 190–192 | MR | Zbl

[14] Zelmanov E. I., Kostrikin A. I., “Teorema o sendvichevykh algebrakh”, Tr. Matem. in-ta AN SSSR, 183, 1988, 106–111 | MR

[15] Iltyakov A. V., “Radikal konechno porozhdennykh iordanovykh PI-algebr”, Sib. matem. zhurn., 26:3 (1985), 44–48 | MR | Zbl

[16] Kuzmin E. N., “Struktura i predstavleniya konechnomernykh algebr Maltseva”, Issledovaniya po teorii kolets i algebr, Tr. In-ta matem. SO AN SSSR, 16, 1989, 75–101 | MR | Zbl

[17] Lvov I. V., Teorema Brauna o radikale konechno porozhdennoi PI-algebry, Preprint No 63, In-t matem. SO AH SSSR, Novosibirsk, 1984

[18] Markov V. T., “O razmernosti nekommutativnykh affinnykh algebr”, Izv. AN SSSR. Ser. matem., 37 (1973), 284–288 | Zbl

[19] Parfenov V. A., “O slabo razreshimom radikale algebr Li”, Sib. matem. zhurn., 12:1 (1971), 171–176 | MR | Zbl

[20] Plotkin B. I., “Ob algebraicheskikh mnozhestvakh elementov v gruppakh i algebrakh Li”, UMN, 13:6 (84) (1958), 133–138 | MR | Zbl

[21] Razmyslov Yu. P., Tozhdestva algebr i ikh predstavlenii, Nauka, M., 1989 | MR

[22] Razmyslov Yu. P., “O radikale Dzhekobsona v PI-algebrakh”, Algebra i logika, 13:3 (1974), 337–360 | MR

[23] Chekhonadskikh A. V., “Nil-idealy iordanovykh troinykh sistem s usloviyami konechnosti”, Sib. matem. zhurn., 26:1 (1985), 159–169 | MR | Zbl

[24] Shestakov I. P., “Absolyutnye deliteli nulya i radikaly konechno porozhdennykh alternativnykh algebr”, Algebra i logika, 15:5 (1976), 585–602 | MR | Zbl

[25] D'Amour A., McCrimmon K., “The local algebras of Jordan systems”, J. Algebra, 177 (1995), 199–239 | DOI | MR | Zbl

[26] Anquela J. A., Cortés T., “Primitive Jordan pairs and triple systems”, J. Algebra, 184 (1996), 632–678 | DOI | MR | Zbl

[27] Anquela J. A., Cortés T., Zelmanov E., “Local nilpotency of the McCrimmon radical of a Jordan system”, Tr. MIAN, 292, 2016, 7–15 | DOI | MR | Zbl

[28] Baxter W. E., Martindale W. S., 3rd., “Central closure of semiprime non-associative rings”, Commun. Algebra, 7:11 (1979), 1103–1132 | DOI | MR | Zbl

[29] Beidar K. I., Martindale W. S., 3rd, Mikhalev A. V., Rings with Generalized Identities, Marcel Dekker, New York, 1996 | MR | Zbl

[30] Erickson T. S., Martindale W. S., 3rd, Osborn J. M., “Prime non-associative algebras”, Pacific J. Math., 60:1 (1975), 49–63 | DOI | MR | Zbl

[31] Fernández López A., Jordan Structures in Lie Algebras, Math. Surv. Monogr. Amer. Math. Soc., 240, Amer. Math. Soc., Providence, RI, 2019 | MR | Zbl

[32] Fernández López A., García E., Gómez Lozano M., “The Jordan algebras of a Lie algebra”, J. Algebra, 308 (2007), 164–177 | DOI | MR | Zbl

[33] Fernández López A., Golubkov A. Yu., “Lie algebras with an algebraic adjoint representation revisited”, Manuscripta Math., 140:3 (2013), 363–376 | MR | Zbl

[34] García E., Gómez Lozano M., “A characterization of the Kostrikin radical of a Lie algebra”, J. Algebra, 346 (2011), 266–283 | DOI | MR | Zbl

[35] García E., Neher E., “Tits-Kantor-Koecher superalgebras of Jordan superpairs covered by grids”, Commun. Algebra, 31:7 (2003), 3335–3375 | DOI | MR | Zbl

[36] Hogben L., McCrimmon K., “Maximal modular inner ideals and the Jacobson radical of a Jordan algebra”, J. Algebra, 68 (1981), 155–169 | DOI | MR | Zbl

[37] Kamiya N., “On the Jacobson radical of infinite-dimensional Lie algebras”, Hiroshima Math. J., 9 (1979), 37–40 | DOI | MR | Zbl

[38] Lister W. G., “Ternary rings”, Trans. Amer. Math. Soc., 154 (1971), 37–55 | DOI | MR | Zbl

[39] Loos O., “Über eine Beziehung zwischen Malcev-Algebren und Lie-Tripelsystemen”, Pacific J. Math., 18:3 (1966), 553–562 | DOI | MR | Zbl

[40] Loos O., “Assoziative Tripelsysteme”, Manuscripta Math., 7:2 (1972), 103–112 | DOI | MR | Zbl

[41] Loos O., Jordan Pairs, Lect. Notes Math., 460, Springer, New York, 1975 | DOI | MR | Zbl

[42] Marcos J. C., Velasco M. V., “The Jacobson radical of a non-associative algebra and the uniqueness of the complete norm topology”, Bull. London Math. Soc., 42:6 (2010), 1010–1020 | DOI | MR | Zbl

[43] McCrimmon K., “The radical of a Jordan algebra”, Proc. Nat. Acad. Sci. USA, 62 (1969), 671–678 | DOI | MR | Zbl

[44] McCrimmon K., A Taste of Jordan Algebras, Springer, New York, 2004 | MR | Zbl

[45] Meyberg K., Lectures on Algebras and Triple Systems, Univ. of Virginia, Charlottesville, 1972 | MR

[46] Myung H. C., “A characterization of the Jacobson radical in ternary algebras”, Proc. Amer. Math. Soc., 38:2 (1973), 228–234 | DOI | MR | Zbl

[47] Nǎstǎsescu C., Van Oystaeyen F., Methods of Graded Rings, Lect. Notes Math., 1836, Springer, Berlin, 2004 | MR

[48] Rowen L. H., “Some results on the center of ring with polynomial identity”, Bull. Amer. Math. Soc., 1973, no. 1, 219–223 | DOI | MR | Zbl

[49] Rowen L. H., Polynomial Identities in Ring Theory, Pure Appl. Math., 84, Academic Press, London, 1980 | MR | Zbl

[50] Thedy A., “Radicals of right-alternative and Jordan rings”, Commun. Algebra, 12:7 (1984), 857–887 | DOI | MR | Zbl

[51] Wisbauer R., Modules and Algebras: Bimodule Structure and Group Actions on Algebras, Pitman Monogr. Surv. Pure Appl. Math., 81, CRC Press, 1996 | MR