Problems of classifying associative or Lie algebras over a field of characteristic not two and finite metabelian groups are wild

Belitskii, Genrich ; Dmytryshyn, Andrii R. ; Lipyanski, Ruvim ; Sergeichuk, Vladimir V. ; Tsurkov, Arkady

The electronic journal of linear algebra, Tome 18 (2009), pp. 516-529.

Résumé

Summary: Let F be a field of characteristic different from 2. It is shown that the problems of classifying (i) local commutative associative algebras over F with zero cube radical, (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and (iii) finite p-groups of exponent p with central commutator subgroup of order p 3 are hopeless since each of them contains $\bullet$the problem of classifying symmetric bilinear mappings U $\times U\to V$ , or $\bullet$the problem of classifying skew-symmetric bilinear mappings U $\times U\to V$ , in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.

Détail

Classification : 15A21, 16G60
Mots-clés : wild problems