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On finite basis of identities of Lie algebra representations. (English) Zbl 0892.17007

Let \(L\) be a Lie algebra over a field \(F\) and let \(A\) be some enveloping algebra of \(L\). A polynomial \(f=f(x_1,\ldots,x_m)\) of the free associative algebra \(F\langle X\rangle\) is called a weak polynomial identity for the pair \((A,L)\) if \(f(g_1,\ldots,g_m)=0\) for all \(g_1,\ldots,g_m\in L\subset A\). For the representation \(\rho:L\to \text{End}_F(V)\) of \(L\) in the vector space \(V\) one defines the identities of \(\rho\) as the weak polynomial identities of the pair \((\text{End}_F(V),\rho(L))\). The weak polynomial identities and the identities of representations are of independent interest for the PI-theory as well as being very useful tools in the study of the “ordinary” polynomial identities of associative and Lie algebras. [See Yu. P. Razmyslov, Identities of algebras and their representations, Transl. Math. Monographs 138, AMS, Providence, RI (1994; Zbl 0827.17001) for more information. ]
After Kemer gave the positive solution of the famous Specht problem showing that the polynomial identities of any associative algebra \(A\) over a field of characteristic 0 follow from a finite number of identities of \(A\) [see A. R. Kemer, Ideals of identities of associative algebras, Transl. Math. Monographs 87, AMS, Providence, RI (1991; Zbl 0732.16001)] for detailed exposition of his theory of T-ideals), there have been different attempts to answer affirmatively the Specht problem for other classes of algebras. For example, [A. Ya. Vais and E. I. Zelmanov, Sov. Math. 33, No. 6, 38-47 (1990); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1989, No. 6, 42-51 (1989; Zbl 0679.17013)] handles the case of finitely generated Jordan algebras.
It is well known that the existence of a polynomial identity of a Lie algebra influences not so much the structure of the algebra as in the case of associative algebras. It is natural to try to transfer the Kemer solution of the Specht problem to Lie algebras which are embedded into associative PI-algebras and this is the purpose of the paper under review.
The main result is the following: If \(L\) is a finitely generated Lie algebra over a field of characteristic 0 and if the enveloping algebra \(A\) of \(L\) is a PI-algebra, then every pair satisfying all weak polynomial identities of \((A,L)\) has a finite basis of its weak identities.
This theorem has a very important consequence. If \(L\) is a finitely generated Lie algebra and \(\text{ Ad}(L)\) is an (associative) PI-algebra, then the variety of Lie algebras generated by \(L\) satisfies the Specht property. In particular, this holds when the Lie algebra \(L\) is finite dimensional.
Reviewer: V.Drensky (Sofia)

MSC:

17B01 Identities, free Lie (super)algebras
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
17B35 Universal enveloping (super)algebras
16R50 Other kinds of identities (generalized polynomial, rational, involution)
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
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