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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.
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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