English version PDF (1727 kB) Citations (7)

DOI: https://doi.org/10.1070/SM1990v067n02ABEH002097

Abstract: 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.

Received: 21.03.1988

Russian version:
Matematicheskii Sbornik, 1989, Volume 180, Number 8, Pages 1033–1066

Bibliographic databases:

Citation: O. G. Kharlampovich, “The word problem for solvable Lie algebras and groups”, Mat. Sb., 180:8 (1989), 1033–1066; Math. USSR-Sb., 67:2 (1990), 489–525

Citation in format AMSBIB

Linking options:

https://www.mathnet.ru/eng/sm1649

https://doi.org/10.1070/SM1990v067n02ABEH002097

https://www.mathnet.ru/eng/sm/v180/i8/p1033

This publication is cited in the following 7 articles:

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles