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Skew-symmetric identities in special Lie algebras. (English. Russian original) Zbl 0855.17002
Sb. Math. 186, No. 1, 65-77 (1995); translation from Mat. Sb. 186, No. 1, 65-78 (1995).
Connections between a standard identity and Capelli identities in Lie algebras over a field of characteristic 0 are investigated. The standard identity of degree \(n\) is the identity \(s_n (x_0, \dots, x_n)\) \(\neq 0\), where \(s_n (x_0, \dots, x_n)\) is \(\sum_{\sigma\in S_n} (-1)^\sigma x_{\sigma (1)} \dots x_{\sigma (n)} x_0\). The system \(C_m\) of Capelli identities of rank \(m\) is the set of all multilinear identities \(f(x_1, \dots, x_k)= 0\) each of them containing a skew-symmetric set of \(m\) variables. Then Theorem 1 states that, in a special Lie algebra over a field of characteristic 0, the identity \(s_7 =0\) implies \(C_m\) for some \(m\) while \(s_8 =0\) does not imply \(C_m\) for any \(m\). In Theorem 2 a skew-symmetric identity equivalent to \(C_m\) in the class of special Lie algebras is discovered. It is known that a special variety of Lie algebras has a finite basis rank if and only if it satisfies \(C_m\) for some \(m\) [see the author, Vestn. Mosk. Univ., Ser. I 1993, No. 1, 56-59 (1993; Zbl 0817.17006)]. According to Theorem 2, a special variety of Lie algebras over a field of characteristic 0 has a finite basis rank if and only if it satisfies an identity of the kind \[ \sum_{\sigma, r\in S_n} (-1)^\sigma x_{\sigma (1)} y_{\tau (1)} \cdots x_{\sigma (n)} y_{\tau (n)} x_0 =0. \] A class of special Lie algebras is contained in the class of so-called Lie API-algebras [see S. P. Mishchenko, Vestn. Mosk. Univ., Ser. I 1992, No. 3, 55-57 (1992; Zbl 0779.17003)]. Let \({\mathcal N}_k\) be the variety of all nilpotent of step \(\leq k\) Lie algebras. Theorem 3 states that if \({\mathcal X}\) is an API-variety and \({\mathcal X} \subseteq {\mathcal N}_q\) for some \(p\), \(q\) then a standard identity implies in \({\mathcal X}\) all Capelli identities of some rank. If \(q\leq 2\) then the conclusion of Theorem 3 remains true without the hypothesis that \({\mathcal X}\) is an API-variety (Theorem 4). Theorem 5 states that, for any \(n\geq 2\), the Lie derivation algebra \(W_n\) satisfies some identity which does not follow from \(s_9 =0\).

MSC:
17B01 Identities, free Lie (super)algebras
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