Finite approximability and representability of Lie algebras (Q1358559)

The Lie algebras considered in this article are over an infinite field. An algebra is called residually finite (or finitely approximable) if it is a subdirect product of finite dimensional algebras. An algebra \(L\) over a field \(F\) is called representable if there is a field extension \(\widetilde{F}\) of \(F\) and a finite dimensional \(\widetilde{F}\)-algebra \(\widetilde{L}\) such that \(L\) is embeddable in \(\widetilde{L}\) as an \(F\)-algebra. It is known that residual finiteness is a necessary condition for representability see \textit{A. I. Mal'tsev} [Mat. Sb. 13, 263-286 (1943; Zbl 0060.07804)] and \textit{S. V. Pchelintsev} [Sib. Mat. Zh. 30, No. 1, 134-144 (1989; Zbl 0671.17019)]). It is known also that a representable Lie algebra is special (i.e., is embeddable in an associative PI-algebra). Theorem 1 shows that these necessary conditions are not sufficient: it gives an example of a residually finite special finitely generated but not representable Lie algebra. Let \({\mathcal V}_n\) be the variety of Lie algebras generated by the matrix algebra \(M_n(F)\). Theorem 1 implies the following Corollary 1: if \(n\geq2\) then, for any \(k\geq n\), there is a finite dimensional Lie algebra \(L\) over a field \(F\) such that \(L\in{\mathcal V}_n\) and \(L\) is not embeddable in \(M_k(Q)\) where \(Q\) is a commutative \(F\)-algebra. Let \(H=H_1\) be a 3-dimensional Heisenberg algebra and \(M=F[t]\) an \(H\)-module with \(xf(t)=df/dt\), \(yf(t)=tf(t)\), \(zf(t)=f(t)\). Theorem 2 states that if \(F\) has a characteristic 0 then the relations \[ [x,y,z]=0,\;\sum\limits_{\sigma\in S_4}x_{\sigma(1)}\cdots x_{\sigma(4)}=0 \] form the identity basis of the representation of \(H\) in \(M\). This theorem implies Corollary 2: if \(W\) is an arbitrary \(H\)-module then all identities of the Lie algebra \(H\leftthreetimes M\) hold in \(H\leftthreetimes W\) (here \(\leftthreetimes\) means the semidirect product). The last result of the article is Theorem 3: if \(\mathcal V\) is a variety of Lie algebras over an infinite field in which all finitely generated residually small algebras are representable then \(\mathcal V\) is locally representable. A complete description of locally representable varieties of Lie algebras is obtained by the author [\textit{M. V. Zaitsev}, Mat. Sb. 180, No. 6, 798-808 (1989; Zbl 0688.17004)]. No proofs are included in the article.