Lie algebras with extremal properties (Q901911)

The author presents two series of varieties of Lie algebras with extremal properties over the field of characteristic zero. The algebras of the first series belong to the Volichenko variety which is of almost polynomial growth. Namely, let \(G_{2k}\), \(k\geq 1\), be the variety of Lie algebras generated by \(2\times 2\) matrices, such that the elements of the first row belong to the Grassmann algebra \(\Lambda_{2k}\) of rank \(2k\), the second row being zero. The author studies properties of these varieties. In particular, he computes the cocharacter sequence and proves that these varieties are of minimal degree polynomial growth, i.e. each proper subvariety is of polynomial growth of smaller degree. Each algebra of the second series generates a variety of polynomial growth minimal with respect to the leading coefficient of the polynomial. The algebras of this series belong to the variety \(N_2A\) of almost polynomial growth.