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An almost nilpotent variety of exponent 2. (English) Zbl 1322.17001

Summary: We construct a nonassociative algebra \(A\) over a field of characteristic zero with the following properties: if \(\mathcal V\) is the variety generated by \(A\), then \(\mathcal V\) has exponential growth but any proper subvariety of \(\mathcal V\) is nilpotent. Moreover, by studying the asymptotics of the sequence of codimensions of \(A\) we deduce that \(\exp(\mathcal V)=2\).

MSC:

17A30 Nonassociative algebras satisfying other identities
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References:

[1] A. Berele and A. Regev, Applications of Hook Young diagrams to P.I. algebras, Journal of Algebra 82 (1983),559–567. · Zbl 0517.16013
[2] N. L. Biggs, Discrete Mathematics, Clarendon Press, Oxford, 1989.
[3] V. Drensky, Free Algebras and PI-Algebras, Graduate Course in Algebra, Springer, Singapore, 2000. · Zbl 0936.16001
[4] A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Advances in Mathematics 140 (1998), 145–155. · Zbl 0920.16012
[5] A. Giambruno and M. Zaicev, Exponential codimension growth of P.I. algebras: an exact estimate, Advances in Mathematics 142 (1999), 221–243. · Zbl 0920.16013
[6] A. Giambruno, S. Mishchenko, and M. Zaicev, Codimensions of algebras and growth functions, Advances in Mathematics 217 (2008), 1027–1052. · Zbl 1133.17001
[7] A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Mathematical Surveys and Monographs, Vol. 122, American Mathematical Society, Providence, RI, 2005. · Zbl 1105.16001
[8] G James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, Vol. 16, Addison-Wesley, London, 1981. · Zbl 0491.20010
[9] S. P. Mishchenko, Varieties of linear algebras with colength one, Moscow University Mathematics Bulletin 65 (2010), 23–27. · Zbl 1304.17002
[10] S. P. Mishchenko and A. Valenti, Varieties with at most quadratic growth, Israel Journal of Mathematics 178 (2010), 209–228. · Zbl 1223.17007
[11] S. Mishchenko and M. Zaicev, An example of a variety of Lie algebras with a fractional exponent, Algebra, 11, Journal of Mathematical Sciences (New York) 93 (1999), 977–982. · Zbl 0933.17004
[12] V. M. Petrogradskii, Growth of polynilpotent varieties of Lie algebras, and rapidly increasing entire functions, Matematicheskiń≠ Sbornik 188 (1997), 119–138; English translation: Sbornik. Mathematics 188 (1997), 913–931.
[13] A. Regev, Existence of identities in A B, Israel Journal of Mathematics 11 (1972), 131–152. · Zbl 0249.16007
[14] M. Zaicev, Integrality of exponents of growth of identities of finite-dimensional Lie algebras, Rossiń≠kaya Akademiya Nauk. Izvestiya. Seriya Matematicheskaya 66 (2002), 23–48; English translation: Izvestiya. Mathematics 66 (2002), 463–487.
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