Varieties of polynomial growth of Lie algebras over a field of characteristic zero.

*(English. Russian original)*Zbl 0624.17007
Math. Notes 40, 901-905 (1986); translation from Mat. Zametki 40, No. 6, 713-722 (1986).

Let \(N_ cA\) be the variety of Lie algebras over a field of characteristic 0 whose derived algebra is nilpotent of class at most c. Let \(P_ n\) be the space of polylinear elements of degree n of the free associative algebra. If M is a Lie algebra variety then let \(T_ n(M)\subset P_ n\) be the subspace of identities of algebras of M. The sequence \(d_ n=d_ n(M)=\dim P_ n/T_ n(M)\) is called the codimension sequence of M.

In his previous article [Vestn. Mosk. Univ., Ser. I 1982, No.5, 63-66 (1982; Zbl 0517.17006)] the author proved that \(M\subset N_ cA\) for some c provided \(d_ n(M)<(\sqrt{2})^{n+2}\) for \(n>n_ 0\), in particular if the growth of \(d_ n(M)\) is subexponential. On the other hand he proved (unpublished) that \(N_ 2A\) is of exponential growth. In the article under review the author proves that M is of polynomial growth if \(M\subset N_ cA\) and \(N_ 2A\not\subset M\). Therefore there is no Lie algebra variety with subexponential and nonpolynomial growth of the codimension sequence.

In his previous article [Vestn. Mosk. Univ., Ser. I 1982, No.5, 63-66 (1982; Zbl 0517.17006)] the author proved that \(M\subset N_ cA\) for some c provided \(d_ n(M)<(\sqrt{2})^{n+2}\) for \(n>n_ 0\), in particular if the growth of \(d_ n(M)\) is subexponential. On the other hand he proved (unpublished) that \(N_ 2A\) is of exponential growth. In the article under review the author proves that M is of polynomial growth if \(M\subset N_ cA\) and \(N_ 2A\not\subset M\). Therefore there is no Lie algebra variety with subexponential and nonpolynomial growth of the codimension sequence.

Reviewer: A.Zalesskij

##### MSC:

17B99 | Lie algebras and Lie superalgebras |

08B15 | Lattices of varieties |

17B30 | Solvable, nilpotent (super)algebras |

17B65 | Infinite-dimensional Lie (super)algebras |

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\textit{S. P. Mishchenko}, Math. Notes 40, 901--905 (1986; Zbl 0624.17007); translation from Mat. Zametki 40, No. 6, 713--722 (1986)

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##### References:

[1] | I. I. Benediktovich and A. E. Zalesskii, ?T-ideals of a free Lie algebra with polynomial growth of the sequence of codimensions,? Vestsi Akad. Nauk BSSR, No. 3, 5-10 (1980). · Zbl 0434.17009 |

[2] | S. P. Mishchenko, ?Varieties of Lie algebras with a weak growth of the sequence of codimensions,? Vestn. Mosk. Gos. Univ., Ser. 1, Mat. Mekh., No. 5, 63-66 (1982). |

[3] | S. P. Mishchenko, ?A variety of Lie algebras whose commutator subalgebra is nilpotent of step two,? Vestn. Mosk. Gos. Univ., Ser. 1, Mat., Mekh., No. 5, 94 (1984). |

[4] | Yu. A. Bakhturin, Identities in Lie Algebras [in Russian], Nauka, Moscow (1985). · Zbl 0571.17001 |

[5] | C. Curtis and I. Reiner, Theory of Representations of Finite Groups and Associative Algebras [Russian translation], Nauka, Moscow (1979). |

[6] | I. B. Volichenko, ?On one variety of Lie algebras associated with standard identities, II,? Vestsi Akad. Nauk BSSR, No. 2, 22-29 (1980). · Zbl 0432.17005 |

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