Growth in Poisson algebras

S. M. Ratseev

Algebra i logika, Tome 50 (2011) no. 1, pp. 68-88.

Résumé

A criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let $V$ be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities $$ \{\{x_1,y_1\},\{x_2,y_2\},\dots,\{x_m,y_m\}\}=0,\qquad\{x_1,y_1\}\cdot\{x_2,y_2\}\cdot\ldots\cdot\{x_m,y_m\}=0, $$ for some $m$. It is shown that the exponent of $V$ exists and is an integer. For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.

Détail

Mots-clés : Poisson algebra
Keywords: growth of variety, colength of variety.

Bibliographie

[1] S. P. Mishchenko, V. M. Petrogradsky, A. Regev, “Poisson PI algebras”, Trans. Am. Math. Soc., 359:10 (2007), 4669–4694 | DOI | MR | Zbl

[2] I. P. Shestakov, “Kvantovaniya superalgebr Puassona i spetsialnost iordanovykh superalgebr puassonova tipa”, Algebra i logika, 32:5 (1993), 571–584 | MR | Zbl

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

[4] V. Drensky, Free algebras and PI-algebras, Graduate course in algebra, Springer-Verlag, Singapore, 2000 | MR | Zbl

[5] D. R. Farkas, “Poisson polynomial identities”, Comm. Algebra, 26:2 (1998), 401–416 | DOI | MR | Zbl

[6] D. R. Farkas, “Poisson polynomial identities. II”, Arch. Math. (Basel), 72:4 (1999), 252–260 | MR | Zbl

[7] S. M. Ratseev, “Rost i kodlina prostranstv spetsialnogo vida mnogoobrazii algebr Puassona”, Izv. vuzov. Povolzh. region, 2006, no. 5(26), 125–135

[8] Yu. N. Maltsev, “Bazis tozhdestv algebry verkhnetreugolnykh matrits”, Algebra i logika, 10:4 (1971), 393–400 | MR

[9] V. M. Petrogradskii, “O chislennykh kharakteristikakh podmnogoobrazii trekh mnogoobrazii algebr Li”, Matem. sb., 190:6 (1999), 111–126 | MR | Zbl

[10] V. M. Petrogradsky, “Exponents of subvarieties of upper triangular matrices over arbitrary fields are integral”, Serdica Math. J., 26:2 (2000), 167–176 | MR | Zbl

[11] S. M. Ratseev, “Rost nekotorykh mnogoobrazii algebr Leibnitsa”, Vest. Samarskogo gos. un-ta, 2006, no. 6(46), 70–77 | MR | Zbl

[12] S. P. Mischenko, “Mnogoobraziya algebr Li so slabym rostom posledovatelnosti korazmernostei”, Vest. MGU, 1982, no. 5, 63–66

[13] S. M. Ratseev, “Rost mnogoobrazii algebr Leibnitsa s nilpotentnym kommutantom”, Matem. zametki, 82:1 (2007), 108–117 | MR

[14] A. R. Kemer, “Shpekhtovost $T$-idealov so stepennym rostom korazmernostei”, Sib. matem. zh., 19:1 (1978), 54–69 | MR | Zbl

[15] I. I. Benediktovich, A. E. Zalesskii, “$T$-idealy svobodnykh algebr Li s polinomialnym rostom posledovatelnosti korazmernostei”, Izv. AN BSSR. Ser. fiz. matem. n., 1980, no. 3, 5–10 | MR | Zbl

[16] S. P. Mischenko, O. I. Cherevatenko, “Mnogoobraziya algebr Leibnitsa slabogo rosta”, Vest. Samarskogo gos. un-ta, 2006, no. 9(49), 19–23

[17] S. P. Mischenko, “O mnogoobraziyakh polinomialnogo rosta algebr Li nad polem kharakteristiki nul”, Matem. zametki, 40:6 (1986), 713–721 | MR | Zbl