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Graded PI-exponents of simple Lie superalgebras. (English) Zbl 1408.17002

By analogy with the case of ordinary polynomial identities for a Lie superalgebra \(A\) one can define the \(\mathbb{Z}_2\)-graded codimension sequence \(c_n^{\operatorname{gr}}(A)\), \(n=1,2,\dots\), and the graded exponent \(\exp^{\operatorname{gr}}(A)=\lim_{n\to\infty} \sqrt[n]{c_n^{\operatorname{gr}}(A)}\) (if this limit exists). The main result of the paper under review is that for any finite dimensional simple Lie superalgebra \(A\) over a field of characteristic zero the graded exponent \(\exp^{\operatorname{gr}}(A)\) exists. A celebrated result in [\textit{A. Giambruno} and \textit{M. Zaicev}, Adv. Math. 142, No. 2, 221--243 (1999; Zbl 0920.16013)] gives that for an associative PI-algebra \(A\) the ordinary exponent \(\exp(A)\) is an integer. The problem for the existing of the ordinary and graded exponents for arbitrary Lie superalgebras is still open. It is also unknown whether both exponents are integer for finite dimensional simple Lie superalgebras. On the other hand, for the seven-dimensional Lie superalgebra \)b(n)\( the results in [\textit{A. Giambruno} and \textit{M. Zaicev}, J. Lond. Math. Soc., II. Ser. 85, No. 2, 534--548 (2012; Zbl 1271.17003)] and [\textit{D. Repovš} and \textit{M. Zaicev}, J. Algebra 422, 1--10 (2015; Zbl 1387.17008)] give, respectively, that \(6<\exp(b(2))<7\) and \(exp^{\operatorname{gr}}(b(2))=3+2\sqrt{3}\)\)

MSC:

17B01 Identities, free Lie (super)algebras
17B20 Simple, semisimple, reductive (super)algebras
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16P90 Growth rate, Gelfand-Kirillov dimension
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References:

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