Universal equivalence of symplectic groups

E. I. Bunina ; A. M. Lazarev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2021) no. 4, pp. 17-38.

Résumé

In this paper, we prove a criterion of universal equivalence of symplectic linear groups over fields: two symplectic linear groups $\mathrm{Sp}_{2n}(K)$ and $\mathrm{Sp}_{2m}(M)$, where $n,m\geq 1$ and $K$ and $M$ are infinite fields of characteristic not equal to $2$, are universally equivalent if and only if $n=m$ and the fields $K$ and $M$ are universally equivalent.

Bibliographie

[1] Al-Shedivat M. F., Bunina E. I., Mikhalev A. V., “Elementarnaya opredelimost lineinykh klassicheskikh i simplekticheskikh grupp nad polyami”, Fundament. i prikl. matem. (to appear)

[2] Bunina E. I., “Elementarnaya ekvivalentnost unitarnykh lineinykh grupp nad koltsami i telami”, UMN, 53:2 (1998), 137–138 | MR | Zbl

[3] Bunina E. I., “Elementarnaya ekvivalentnost unitarnykh lineinykh grupp nad polyami”, Fundament. i prikl. matem., 4:4 (1998), 1265–1278 | MR | Zbl

[4] Bunina E. I., “Elementarnaya ekvivalentnost grupp Shevalle”, UMN, 56:1 (2001), 157–158 | MR | Zbl

[5] Bunina E. I., “Elementarnaya ekvivalentnost grupp Shevalle nad lokalnymi koltsami”, Matem. sb., 201:3 (2010), 3–20 | MR | Zbl

[6] Bunina E. I., Kaleeva G. A., “Universalnaya ekvivalentnost obschikh i spetsialnykh lineinykh grupp nad polyami”, Fundament. i prikl. matem., 21:3 (2016), 73–106

[7] Bunina E. I., Mikhalev A. V., Pinus A. G., Elementarnaya i blizkie k nei logicheskie ekvivalentnosti klassicheskikh i universalnykh algebr, MTsNMO, M., 2015

[8] Gurevich Yu. Sh., Kokorin A. I., “Universalnaya ekvivalentnost uporyadochennykh abelevykh grupp”, Algebra i logika, 2:1 (1963), 37–39 | MR | Zbl

[9] Kaleeva G. A., “Universalnaya ekvivalentnost lineinykh grupp nad lokalnymi kommutativnymi koltsami s $1/2$”, Algebra i logika, 58:4 (2019), 467–478 | MR | Zbl

[10] Maltsev A. I., “Ob elementarnykh svoistvakh lineinykh grupp”, Problemy matematiki i mekhaniki, 1961, 110–132 | Zbl

[11] O'Mira O., Lektsii o simplekticheskikh gruppakh, Mir, M., 1979

[12] Taimanov A. D., “Kharakteristiki aksiomatiziruemykh klassov modelei”, Algebra i logika, 1:4 (1962), 5–31 | Zbl

[13] Timoshenko E. I., “Ob universalno ekvivalentnykh razreshimykh gruppakh”, Algebra i logika, 39:2 (2000), 227–240 | MR | Zbl

[14] Khisamiev N. G., “Universalnaya teoriya strukturno uporyadochennykh abelevykh grupp”, Algebra i logika, 5:3 (1966), 71–76 | Zbl

[15] Beidar C. I., Mikhalev A. V., “On Malcev's theorem on elementary equivalence of linear groups”, Contemp. Math., 131, 1992, 29–35 | MR | Zbl

[16] Eklof P. C., “Some model theory for Abelian groups”, J. Symb. Logic, 37 (1972), 335–342 | MR | Zbl

[17] Mishchenko A. A., Timoshenko E. I., “Universal equivalence of partially commutative nilpotent groups”, Sib. Math. J., 52:5 (2011), 884–891 | MR | Zbl

[18] Poroshenko E. N., Timoshenko E. I., “Universal equivalence of partially commutative metabelian Lie algebras”, J. Algebra, 384 (2013), 143–168 | MR | Zbl