an:04012849
Zbl 0624.17007
Mishchenko, S. P.
Varieties of polynomial growth of Lie algebras over a field of characteristic zero
EN
Math. Notes 40, 901-905 (1986); translation from Mat. Zametki 40, No. 6, 713-722 (1986).
00153333
1986
j
17B99 08B15 17B30 17B65
variety of Lie algebras; codimension sequence; polynomial growth
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 On the other hand he proved (unpublished) that \(N_ 2A\) is of exponential growth.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.
A.Zalesskij
Zbl 0517.17006