Abstract
It follows from the Golod–Shafarevich theorem that if $k\in{\mathbb N}$ and $R$ is an associative algebra given by $n$ generators and $d<n^2\cos^{-2}(\pi/(k+1))/4$ quadratic relations, then $R$ is not $k$-step nilpotent. We show that the above estimate is asymptotically optimal. Namely, for every $k$ there is a sequence of algebras $R_n$ given by $n$ generators and $d_n$ quadraticrelations such that $R_n$ is k-step nilpotent and $\lim_{n\to\infty d_n/n^2=\cos^{-2}(\pi/(k+1))/4$.
Original language | English |
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Pages (from-to) | 465–479 |
Number of pages | 15 |
Journal | Combinatorica |
Volume | 37 |
Issue number | 3 |
Early online date | 17 Oct 2016 |
DOIs | |
Publication status | Published - Jun 2017 |