On varieties of Lie algebras and varieties of restricted Lie algebras
Let \(K\) be a field of positive characteristic \(p\), let \(L(X)\) be the free restricted Lie algebra over \(K\) freely generated by a countable set \(X\) and let \(L(X)\) be the free Lie algebra canonically embedded into \(L(X)\). The author constructs a natural correspondence between varieties of restricted Lie algebras \(U\) and varieties of Lie algebras \(U\) as follows: \(U= \tau(U)\) is determined by all Lie polynomial identities in \(U\) and \(U= \eta (U)\) consists of all restricted Lie algebras satisfying the same Lie identities as \(U\). The properties of the mappings \(\tau\) and \(\eta\) are investigated. As a consequence the author establishes that over a finite field \(K\) there exist two solvable just-non-Cross varieties of restricted Lie algebras only, i.e. these varieties are not generated by a finite algebra and all their proper subvarieties are generated by such an algebra. An analogue for restricted Lie algebras of a result of E. V. Ritter on the skeleton of locally finite varieties of Lie algebras is obtained as well. Reviewer's remark. In the title of the English translation of the paper under review, ``bounded Lie algebras'' should be replaced by ``restricted Lie algebras''.
scientific article; zbMATH DE number 848911