An application of Dilworth's Theorem to a problem on free LIE-algebras
The paper is published: Rostocker Mathematische Kolloquium, Rostock. Math. Kolloq. 49, 25-30 (1995)
MSC:
20C20 Modular representations and characters
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
Abstract: A generalization of a theorem of Golod, Safarevic [3] after an idea by
Koch [5] gives the following lower bound for the number r of relations
given by sums of Lie-monoms of defgree m which are necessary to obtain
a nilpotent LIE-algebra as a factor of a free LIE-algebra with d
generators [8]:
\[ \frac{(m-1)^{m-1}}{m^m}d^m In order to study the sharpness of this bound one has to construct
nilpotent LIE-algebras with a small number of relations. Wisliceny [9]
described a class of systems of relations, called increasing systems
(Erhöhungssysteme), which yield nilpotent LIE-algebras. This was
first done for the case m=2 [9] and later this was generalized by him [10]
and Sauerbier [7], [8] to any m\ge 2.
Notes: Abstract contains the first few lines of text of the paper.