Integrality of exponents of codimension growth of finite-dimensional Lie algebras

M. V. Zaicev

Izvestiya. Mathematics , Tome 66 (2002) no. 3, pp. 463-487.

Résumé

We study the asymptotic behaviour of the codimension growth sequence $c_n(L)$ of a finite-dimensional Lie algebra $L$ over a field of characteristic zero. It is known that the growth of the sequence $\{c_n(L)\}$ is bounded by an exponential function of $n$, and hence there exist the upper and lower limits of the $n$th roots of $c_n(L)$, which are called the upper and lower exponents. By Amitsur's conjecture, the upper and lower exponents should coincide and be integers. This conjecture has been confirmed in the associative case for any PI-algebra. For finite-dimensional Lie algebras, a positive solution has been found for soluble, simple and semisimple algebras and also for algebras whose soluble radical is nilpotent. For infinite-dimensional Lie algebras, the problem has been solved in the negative. In this paper we give a proof of Amitsur's conjecture for arbitrary finite-dimensional Lie algebras.

Bibliographie

[1] Mishchenko S. P., Petrogradsky V. M., “Exponents of varieties of Lie algebras with a nilpotent commutator subalgebra”, Commun. Algebra, 27 (1999), 2223–2230 | DOI | MR | Zbl

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

[3] Giambruno A., Regev A., Zaicev M. V., “Simple and semisimple Lie Algebras and codimension growth”, Trans. Amer. Math. Soc., 352 (2000), 1935–1946 | DOI | MR | Zbl

[4] Zaitsev M. V., “Tozhdestva affinnykh algebr Katsa–Mudi”, Vestn. MGU. Ser. matem., mekhan., 1996, no. 2, 33–36 | MR

[5] Zaitsev M. V., “Mnogoobraziya affinnykh algebr Katsa–Mudi”, Matem. zametki, 62:1 (1997), 95–102 | MR | Zbl

[6] Mischenko S. P., “K probleme engelevosti”, Matem. sb., 124:1 (1984), 56–67 | MR | Zbl

[7] Volichenko I. B., “O mnogoobrazii algebr Li $AN_2$ nad polem nulevoi kharakteristiki”, DAN BSSR, 26:12 (1981), 1063–1066 | MR

[8] Petrogradskii V. M., “O tipakh sverkheksponentsialnogo rosta tozhdestv v PI-algebrakh Li”, Fundamentalnaya i prikl. matematika, 1:4 (1995), 989–1007 | MR | Zbl

[9] Petrogradskii V. M., “Rost polinilpotentnykh mnogoobrazii algebr Li i bystro rastuschie tselye funktsii”, Matem. sb., 188:6 (1997), 119–138 | MR | Zbl

[10] Regev A., “Existence of identities in $A \otimes B$”, Israel J. Math., 11 (1972), 131–152 | DOI | MR | Zbl

[11] Giambruno A., Zaicev M. V., “On codimension growth of finitely generated associative algebras”, Advances in Mathematics, 140 (1998), 145–155 | DOI | MR | Zbl

[12] Giambruno A., Zaicev M. V., “Exponential codimension growth of P.I. algebras: an exact estimate”, Advances in Mathematics, 142 (1999), 221–243 | DOI | MR | Zbl

[13] Giambruno A., Regev A., Zaicev M., “On the codimension growth of finite-dimensional Lie algebras”, J. of Algebra, 220 (1999), 466–474 | DOI | MR | Zbl

[14] Zaicev M. V., Mishchenko S. P., “An example of a variety of Lie algebras with a fractional exponent”, J. of Mathematical Sciences, 93:6 (1999), 977–982 | DOI | MR | Zbl

[15] Zaitsev M. V., Mischenko S. P., “Kriterii polinomialnosti rosta mnogoobrazii superalgebr Li”, Izv. RAN. Ser. matem., 62:5 (1998), 103–116 | MR | Zbl

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

[17] Goto M., Grosskhans F., Poluprostye algebry Li, Mir, M., 1981 | MR | Zbl

[18] Kaplanskii I., Algebry Li i lokalno kompaktnye gruppy, Mir, M., 1974

[19] Mischenko S. P., “Nizhnie otsenki razmernostei neprivodimykh predstavlenii simmetricheskikh grupp i pokazatelei eksponenty mnogoobrazii algebr Li”, Matem. sb., 187:1 (1996), 83–94 | MR | Zbl