The word problem for solvable Lie~algebras and groups

O. G. Kharlampovich

Sbornik. Mathematics, Tome 67 (1990) no. 2, pp. 489-525

Résumé

The variety of groups $Z\mathfrak N_2\mathfrak A$ is given by the identity $$ [[x_1,x_2],[x_3,x_4],[x_5,x_6],x_7]=1, $$ and the analogous variety of Lie algebras is given by the identity $$ (x_1x_2)(x_3x_4)(x_5x_6)x_7=0. $$ Previously the author proved the unsolvability of the word problem for any variety of groups (respectively: Lie algebras) containing $Z\mathfrak N_2\mathfrak A$, and its solvability for any subvariety of $\mathfrak N_2\mathfrak A$. Here the word problem is investigated in varieties of Lie algebras over a field of characteristic zero and in varieties of groups contained in $Z\mathfrak N_2\mathfrak A$. It is proved that in the lattice of subvarieties of $Z\mathfrak N_2\mathfrak A$ there exist arbitrary long chains in which the varieties with solvable and unsolvable word problems alternate. In particular, the variety $Z\mathfrak N_2\mathfrak A\frown\mathfrak N_2\mathfrak N_c$ has a solvable word problem for any $c$, while the variety $\mathfrak Y_2$, given within $Z\mathfrak N_2\mathfrak A$ by the identity $$ [[x_1,\dots,x_{2c+2}],[y_1,\dots,y_{2c+2}],[z_1,\dots,z_{2c}]]=1 $$ in the case of groups and by the identity $$ (x_1\dots x_{2c+2})(y_1\dots y_{2c+2})(z_1\dots z_{2c})=0 $$ in the case of Lie algebras, has an unsolvable word problem. It is also proved that in $Z\mathfrak N_2\mathfrak A$ there exists an infinite series of minimal varieties with an unsolvable word problem, i.e. varieties whose proper subvarieties all have solvable word problems. Bibliography: 17 titles.