Algebraic Lie algebras with finite grading

A. Yu. Golubkov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 1, pp. 87-102.

Résumé

The paper presents a variant of the proof of local finite-dimensionality of Lie PI-algebras with an algebraic adjoint representation over fields of characteristic zero without the use of extremal elements, a number of similar conclusions for such algebras over fields of characteristic $p> 7$, and generalizes the description of the locally finite radical of algebraic Mal'tsev locally PI-algebras to any base field of characteristic zero.

Bibliographie

[1] Bakhturin Yu. A., Tozhdestva v algebrakh Li, Nauka, M., 1985 | MR

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

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

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

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

[6] Golubkov A. Yu., “Iordanova algebra algebry Maltseva”, Fundament. i prikl. matem., 23:3 (2020), 49–74

[7] Golubkov A. Yu., “Kvaziregulyarnye radikaly neassotsiativnykh algebr”, Fundament. i prikl. matem., 24:4 (2023), 75–128 | MR

[8] Golubkov A. Yu., Lokalno konechnyi radikal algebraicheskikh algebr Maltseva, V pechati

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

[10] Dzhekobson N., Algebry Li, Mir, M., 1964

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

[12] Zhevlakov K. A., Shestakov I. P., “O lokalnoi konechnosti v smysle Shirshova”, Algebra i logika, 12:1 (1973), 43–73 | MR

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

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

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

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

[17] Zelmanov E. I., “O pervichnykh iordanovykh troinykh sistemakh. III”, Sib. matem. zhurn., 26:1 (1985), 71–82 | MR | Zbl

[18] Kostrikin A. I., Vokrug Bernsaida, Nauka, M., 1986 | MR

[19] Kuzmin E. N., “Algebraicheskie mnozhestva v algebrakh Maltseva”, Algebra i logika, 7:2 (1968), 42–47 | 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] Premet A. A., “Algebry Li bez silnogo vyrozhdeniya”, Matem. sb., 129 (171):1 (1983), 140–153

[22] Premet A. A., “Vnutrennie idealy modulyarnykh algebr Li”, Vestsi AN BSSR. Ser. fiz.-mat. navuk, 1986, no. 5, 11–15 | Zbl

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

[24] Amitsur S. A., “Identities in rings with involutions”, Israel J. Math., 7:1 (1969), 63–68 | DOI | 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., Montaner F., “The structure of primitive quadratic Jordan algebras”, J. Algebra, 172 (1995), 530–553 | 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] Cabrera M., Fernández López A., Golubkov A. Yu., Moreno A., “Algebras whose multiplication algebra is PI or GPI”, J. Algebra, 2016, no. 459, 213–237 | DOI | MR | Zbl

[31] Draper Fontanals C., Fernández López A., García E., Gómez Lozano M., “The socle of a nondegenerate Lie algebra”, J. Algebra, 319 (2008), 2372–2394 | DOI | MR | Zbl

[32] 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

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

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

[35] Fernández López A., Golubkov A. Yu., “Lie algebras with an algebraic adjoint representation revisited”, Manuscripta Math., 140:3-4 (2013), 363–376 | DOI | MR | Zbl

[36] Henning J., “Simple, locally finite dimensional Lie algebras in positive characteristic”, J. Algebra, 413 (2014), 270–288 | DOI | MR

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

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

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

[40] Premet A., Strade H., “Classification of finite dimensional Lie algebras in prime characteristics”, Representations of Algebraic Groups, Quantum Groups and Lie Algebras, Providence, RI, 413, Amer. Math. Soc., 2006, 185–214 | MR | Zbl

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