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Standard identity and Capelli identities in Lie algebras. (English. Russian original) Zbl 0840.17006
Russ. Math. Surv. 49, No. 2, 186-187 (1994); translation from Usp. Mat. Nauk 49, No. 2(296), 153-154 (1994).
The note under review discusses the impact of the standard Lie identities $$S_n$$ on the Capelli identities in certain classes of Lie algebras over a field of characteristic zero. It is shown that the identity $$s_7= 0$$ implies all Capelli identities $$C_m$$ of a given order $$m$$ in the class of all special Lie algebras. In addition the identity $$s_8 =0$$ does not imply any $$C_m$$ in this class of Lie algebras. The class of all API Lie algebras is considered, too. (Recall that the Lie algebra $$L$$ is called an API algebra if it satisfies some identity of the form $\sum (-1)^s (-1)^t x_{s(1)} y_{t(1)} \dots x_{s(n)} y_{t(n)} x=0$ where the summation is over all permutations $$s$$, $$t$$ of $$1, 2, \dots,n$$, and $$(-1)^s$$ stands for the sign of the permutation $$s$$.) Namely it is proved that if $$L$$ is an API algebra then $$L$$ satisfies all Capelli identities of a given degree when $$L$$ satisfies some standard identity, and $$L$$ lies in the product of two nilpotent varieties of Lie algebras.
MSC:
 17B01 Identities, free Lie (super)algebras 16R99 Rings with polynomial identity 17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
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