Construction of finitely graded associative algebras
The author constructs associative finitely graded (nilpotent) algebras with a small number of relations. By the theorem of Golod and Shafarevich \(r>((m-1)^{m-1}/m^ m) d^ m\), where \(r\) is the number of relations, \(d\) the number of generators and \(m\) is the height of the algebra (``Stufe''). In an earlier paper it was shown for \(m=2\) that \(r/d^ 2\) can be arbitrarily close to 1/4, for \(m=3\) likewise \(r/d^ 3\) can be arbitrarily close to \(\sqrt 3/3\). In this paper the statement is generalized to the following (Satz 2). There is a family of algebras of height \(m\) and number of generators \(d\) such that \[ \lim_{d\to\infty}{r(d)\over d^ m}={1\over m}\root m-1\of m. \] {}.
scientific article; zbMATH DE number 95950