Burnside-type problems, theorems on height, and independence

A. Ya. Belov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 5, pp. 19-79.

Résumé

This review paper is devoted to some questions related to investigations of bases in PI-algebras. The central point is generalization and refinement of the Shirshov height theorem, of the Amitsur–Shestakov hypothesis and of the independence theorem. The paper is mainly inspired by the fact that these topics shed some light on the analogy between structure theory and constructive combinatorial reasoning related to the “microlevel,” to relations in algebras and straightforward calculations. Together with the representation theory of monomial algebras, height and independence theorems are closely connected with combinatorics of words and of normal forms, as well as with properties of primary algebras and with combinatorics of matrix units. Another subject of this paper is an attempt to create a kind of symbolic calculus of operators defined on records of transformations.

Bibliographie

[1] Beidar K. I., Latyshev V. N., Markov V. T., Mikhalëv A. V., Skornyakov L. A., Tuganbaev A. A., “Assotsiativnye koltsa”, Itogi nauki i tekhniki. Algebra. Topologiya. Geometriya, 22, VINITI, M., 1984, 3–115 | MR

[2] Belov A. Ya., “O bazise Shirshova otnositelno svobodnykh algebr slozhnosti $n$”, Mat. sb., 135:3 (1988), 373–384 | MR | Zbl

[3] Belov A. Ya., “Teorema Nagaty–Khigmana dlya polukolets”, Fundament. i prikl. mat., 1:2 (1995), 523–527 | MR | Zbl

[4] Belov A. Ya., Borisenko V. V., Latyshev V. N., “Monomialnye algebry”, Algebra – 4, Itogi nauki i tekhn. Ser. Sovremennaya matematika i eë prilozheniya, 26, VINITI, M., 2002, 35–214

[5] Grishin A. V., “Pokazatel rosta mnogoobraziya algebr i ego prilozheniya”, Algebra i logika, 28:5 (1987), 536–557 | MR

[6] Dnestrovskaya tetrad, 4-e izd., Izd. In-ta matem. SO AN SSSR, Novosibirsk, 1993 | MR

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

[8] Zelmanov E. I., “Absolyutnye deliteli nulya i algebraicheskie iordanovy algebry”, Sib. mat. zhurn., 23:6 (1982), 100–116 | MR

[9] Zubrilin K. A., “Algebry, udovletvoryayuschie tozhdestvam Kapelli”, Mat. sb., 186:3 (1995), 53–64 | MR | Zbl

[10] Zubrilin K. A., “O klasse nilpotentnosti prepyatstviya dlya predstavimosti algebr, udovletvoryayuschikh tozhdestvam Kapelli”, Fundament. i prikl. mat., 1:2 (1995), 409–430 | MR | Zbl

[11] Kemer A. R., Nematrichnye mnogoobraziya, mnogoobraziya so stepennym rostom i konechno porozhdënnye PI-algebry, Dis. ...kand. fiz.-mat. nauk, Novosibirsk, 1981 | Zbl

[12] Kemer A. R., “Mnogoobraziya i $\mathbb Z_2$-graduirovannye algebry”, Izv. AN SSSR. Ser. mat., 48:5 (1984), 1042–1059 | MR

[13] Kolotov A. T., “O verkhnei otsenke vysoty v konechno porozhdënnykh algebrakh s tozhdestvami”, Sib. mat. zhurn., 23:1 (1982), 187–189 | MR | Zbl

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

[15] Kourovskaya tetrad: Nereshënnye voprosy teorii grupp, 13-e izd., Izd-vo in-ta matem. SO AN SSSR, Novosibirsk, 1995

[16] Kurosh A. G., “Problemy teorii kolets, svyazannye s problemoi Bernsaida o periodicheskikh gruppakh”, Izv. AN SSSR. Ser. mat., 5 (1941), 233–240 | Zbl

[17] Lvov I. V., “K teoreme Shirshova o vysote”, Vsesoyuzn. simp. po teorii kolets, algebr i modulei, NGU, Novosibirsk, 1982

[18] Medvedev Yu. A., Nil-radikaly konechno porozhdënnykh iordanovykh PI-algebr, Preprint No 24, IM SO AN SSSR, Novosibirsk, 1985 | MR

[19] Mischenko S. P., “Variant teoremy o vysote dlya algebr Li”, Mat. zametki, 47:4 (1990), 83–89 | MR

[20] Pchelintsev S. V., “Teorema o vysote dlya alternativnykh algebr”, Mat. sb., 124:4 (1984), 557–567 | MR | Zbl

[21] Razmyslov Yu. P., “Algebry, udovletvoryayuschie tozhdestvennym sootnosheniyam tipa Kapelli”, Izv. AN SSSR. Ser. mat., 45:1 (1981), 143–166 | MR | Zbl

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

[23] Skosyrskii V. G., Iordanovy algebry s usloviem minimalnosti dlya dvustoronnikh idealov, Preprint No 21, IM SO AN SSSR, Novosibirsk, 1985 | MR

[24] Ufnarovskii V. A., “Teorema o nezavisimosti i eë sledstviya”, Mat. sb., 128:1 (1985), 124–132 | MR | Zbl

[25] Ufnarovskii V. A., “Kombinatornye i asimptoticheskie metody v algebre”, Algebra – 6, Itogi nauki i tekhn. Ser. Sovr. probl. matematiki. Fundamentalnye napravleniya, 57, VINITI, M., 1990, 5–177 | MR

[26] Kherstein I., Nekommutativnye koltsa, Mir, M., 1972 | MR

[27] Chekanu G. P., Lokalno konechnye algebry, Dis. ...kand. fiz.-mat. nauk, Kishinëv, 1982 | Zbl

[28] Chekanu G. P., “K teoreme Shirshova o vysote”, XIX Vsesoyuzn. algebr. konf. Tezisy soobsch. I, Lvov, 1987, 306

[29] Chekanu G. P., “O lokalnoi konechnosti algebr”, Mat. issledovaniya, 1988, no. 105, 153–171 | MR | Zbl

[30] Chekanu G. P., “Nezavisimost i kvaziregulyarnost v algebrakh”, 3 Mezhdunar. konf. po algebre pamyati M. I. Kargapolova, Krasnoyarsk, 1993 | Zbl

[31] Chekanu G. P., “Nezavisimost i kvaziregulyarnost v algebrakh”, Dokl. RAN, 337:3 (1994), 316–319 | MR | Zbl

[32] Chekanu G. P., Nezavisimost i lokalnaya konechnost, Dis. ...khabilitat fiz.-mat. nauk, Kishinëv, 1995

[33] Shestakov I. P., “Konechno porozhdënnye spetsialnye iordanovy i alternativnye PI-algebry”, Mat. sb., 122:1 (1983), 31–40 | MR | Zbl

[34] Shirshov A. I., “O nekotorykh neassotsiativnykh nil-koltsakh i algebraicheskikh algebrakh”, Mat. sb., 41:3 (1957), 381–394 | Zbl

[35] Shirshov A. I., “O koltsakh s tozhdestvennymi sootnosheniyami”, Mat. sb., 43:2 (1957), 277–283 | Zbl

[36] Amitsur S., “A generalization of Hilbert's Nullstellensatz”, Proc. Amer. Math. Soc., 8:4 (1957), 649–656 | DOI | MR | Zbl

[37] Amitsur S., “Rational identities and applications to algebra and geometry”, J. Algebra, 3:3 (1966), 304–359 | DOI | MR | Zbl

[38] Amitsur S., “The sequence of codimensions of PI-algebras”, Israel J. Math., 47:1 (1984), 1–22 | DOI | MR | Zbl

[39] Belov A., “Some estimations for nilpotence of nil-algebras over field of an arbitrary characteristics and height theorem”, Comm. Algebra, 20:10 (1992), 2919–2922 | DOI | MR | Zbl

[40] Belov A., “About height theorem”, Comm. Algebra, 23:9 (1995), 3551–3553 | DOI | MR | Zbl

[41] Berele A., “Homogeneous polynomial identities”, Israel J. Math., 42:3 (1982), 258–272 | DOI | MR | Zbl

[42] Bokut' L. A., Shestakov I. P., “Some results by A. I. Shirshov and his school”, Proc. Second Int. Conf. on Algebra Dedicated to the Memory of A. I. Shirshov (August 20–25, 1991, Barnaul, Russia), Contemp. Math., 184, ed. L. A. Bokut', Amer. Math. Soc., Providence, 1995, 1–12 | MR | Zbl

[43] Braun A., “The radical in a finitely generated PI-algebra”, Bull. Amer. Math. Soc., 7:2 (1982), 385–386 | DOI | MR | Zbl

[44] Drensky V., “On the Hilbert series of relatively free algebras”, Comm. Algebra, 12:19 (1984), 2335–2347 | DOI | MR | Zbl

[45] Drensky V., “Codimensions of $T$-ideals and Hilbert series of relatively free algebras”, J. Algebra, 91:1 (1984), 1–17 | DOI | MR | Zbl

[46] Drensky V., Free Algebras and PI-Algebras, Springer, Berlin, 2000 | MR | Zbl

[47] Formanek E., Halpin P. W., Li C. W., “The Poincare series of the ring of $2\times2$ generic matrices”, J. Algebra, 69 (1981), 105–112 | DOI | MR | Zbl

[48] Giambruno A., Zaicev M., “Exponential codimension growth of PI algebras: An exact estimate”, Adv. Math., 142 (1999), 221–243 | DOI | MR | Zbl

[49] Giambruno A., Zaicev M., “Minimal varieties of algebras of exponential growth”, Electron. Res. Announc. Amer. Math. Soc., 6 (2000), 40–44 | DOI | MR | Zbl

[50] Procesi C., Rings with Polynomial Identities, Marcel Dekker, New York, 1973 | MR | Zbl

[51] Rowen L. H., Polynomial Identities in Ring Theory, Academic Press, New York, 1980 | MR | Zbl

[52] Small L. W., “Localization in PI-rings”, J. Algebra, 19:2 (1971), 181–183 | MR

[53] Specht W., “Gesetze in Ringen”, Math. Z., 52 (1950), 557–589 | DOI | MR | Zbl