THE WORD PROBLEM FOR SOLVABLE LIE-ALGEBRAS AND GROUPS

The variety of groups is given by the identity and the analogous variety of Lie algebras is given by the identity Previously the author proved the unsolvability of the word problem for any variety of groups (respectively: Lie algebras) containing , and its solvability for any subvariety of . Here the word problem is investigated in varieties of Lie algebras over a field of characteristic zero and in varieties of groups contained in. It is proved that in the lattice of subvarieties of there exist arbitrary long chains in which the varieties with solvable and unsolvable word problems alternate. In particular, the variety has a solvable word problem for any , while the variety , given within by the identity in the case of groups and by the identity in the case of Lie algebras, has an unsolvable word problem. It is also proved that in there exists an infinite series of minimal varieties with an unsolvable word problem, i.e. varieties whose proper subvarieties all have solvable word problems. © 1990 American Mathematical Society.