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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
##### Citations:
Zbl 0817.17006; Zbl 0779.17003
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