Asymptotically optimal k-step nilpotency of quadratic algebras and the Fibonacci numbers

Abstract

It follows from the Golod-Shafarevich theorem that if k ∈ N and R is an associative algebra given by n generators and

$$d< \frac{{{n^2}}}{4}{\cos ^{ - 2}}\left( {\frac{\pi }{{k + 1}}} \right)$$

quadratic relations, then R is not k-step nilpotent. We show that the above estimate is asymptotically optimal. Namely, for every k ∈ N, there is a sequence of algebras Rn given by n generators and dn quadratic relations such that R _n is k-step nilpotent and

$$\mathop {\lim }\limits_{n \to \infty } \frac{{{d_n}}}{{{n^2}}} = \frac{1}{4}{\cos ^{ - 2}}\left( {\frac{\pi }{{k + 1}}} \right)$$

.