Geodesic
The base rank of varieties of Lie algebras
Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 15-31 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this article it is proved that over a field of characteristic zero the product $V_1,\dots,V_n$ of varieties of Lie algebras in which $V_n$ is nilpotent has, as a rule, infinite base rank. An exception is the case when $n=2$, $ V_2$ is abelian, and $V_1$ is nilpotent. It is also shown that if $V_1$ is abelian and $V_2=\operatorname{var\,sl}_2$, then the base rank of $V_1V_2$ is equal to two. A criterion is obtained for the finiteness of the base rank of a special variety. All special varieties of Lie algebras of almost finite base rank are described. 
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M. V. Zaicev. The base rank of varieties of Lie algebras. Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 15-31. http://geodesic.mathdoc.fr/item/SM_1995_80_1_a1/

[1] Bakhturin Yu. A., Tozhdestva v algebrakh Li, Nauka, M., 1986 | MR

[2] Bahturin Y., “On identical relations in free polynilpotent algebras”, J. London Math. Soc., 20 (1980), 39–52 | DOI | MR

[3] Volichenko I. B., “Ob odnom mnogoobrazii algebr Li, svyazannom so standartnymi tozhdestvami, I”, Izv. AN BSSR. Ser. fiz.-mat. nauk, 1980, no. 1, 23–30 | MR | Zbl

[4] Volichenko I. B., “Ob odnom mnogoobrazii algebr Li, svyazannom so standartnymi tozhdestvami, II”, Izv. AN BSSR. Ser. fiz.-mat. nauk, 1980, no. 2, 22–29 | MR

[5] Zaitsev M. V., “Standartnoe tozhdestvo v spetsialnykh mnogoobraziyakh algebr Li”, Vestn. MGU. Ser. 1. Matematika, mekhanika, 1993, no. 1, 53–56 | MR

[6] Vais A. Ya., “O spetsialnykh mnogoobraziyakh algebr Li”, Algebra i logika, 28 (1989), 29–40 | MR | Zbl

[7] Neiman Kh., Mnogoobraziya grupp, Mir, M., 1969 | MR

[8] Razmyslov Yu. P., “O konechnoi baziruemosti tozhdestv matrichnoi algebry vtorogo poryadka nad polem kharakteristiki nul”, Algebra i logika, 12 (1973), 83–113 | MR | Zbl

[9] Shmelkin A. L., “Spleteniya algebr Li i ikh primenenie v teorii grupp”, Tr. MMO, 29, URSS, M., 1973, 247–260 | MR | Zbl

[10] Mischenko S. P., “Rost mnogoobrazii algebr Li”, UMN, 45 (1990), 25–45 | Zbl

[11] Scheunert M., The theory of Lie superalgebras, Lect. Notes in Math., 716, Sringer-Verlag, 1979 | MR

[12] Scheunert M., Nahm W., Rittenberg V., “Classification of all simple graded Lie algebras whose Lie algebra is reductive, I”, J. Math. Phys., 17 (1976), 1626–1639 | DOI | MR | Zbl

[13] Scheunert M., Nahm W., Rittenberg V., “Classification of all simple graded Lie algebras whose Lie algebra is reductive, II”, J. Math. Phys., 17 (1976), 1640–1644 | DOI | MR | Zbl

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

[15] Maltsev A. I., “O predstavlenii beskonechnykh algebr”, Matem. sb., 13 (1943), 263–286 | MR | Zbl