Locally representable varieties of Lie~algebras

M. V. Zaicev

Sbornik. Mathematics, Tome 67 (1990) no. 1, pp. 249-259

Résumé

A description is obtained for locally representable varieties of Lie algebras, i.e., varieties in which an arbitrary finitely generated algebra has a faithful representation of finite dimension over an extension of the ground field. In the case of an infinite field $\Phi$ a variety $V$ of Lie algebras is locally representable if and only if the following two conditions hold: 1) $zy^nx=\sum\limits_{j=1}^n\alpha_jy^jzy^{n-j}x$ is an identity in $V$ for some $\alpha_1,\dots,\alpha_n$ in $\Phi$; and 2) an arbitrary finitely generated algebra in $V$ lies in a product $N_cN_d$ of nilpotent varieties, where $d=1$ if $\operatorname{char}\Phi=0$. Bibliography: 13 titles.