Optimal 5-step nilpotent quadratic algebras

Stanislav Shkarin, Natalia Iyudu

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


By the Golod–Shafarevich theorem, an associative algebra $R$ given by $n$ generators and $<n^2/3$ homogeneous quadratic relations is not 5-step nilpotent. We prove that this estimate is optimal. Namely, we show that for every positive integer $n$, there is an algebra $R$ given by $n$ generators and $\lceil n^2/3\rceil$ homogeneous quadratic relations such that $R$ is 5-step nilpotent.
Original languageEnglish
Pages (from-to)1-14
Number of pages14
JournalJournal of Algebra
Early online date17 May 2014
Publication statusPublished - 15 Aug 2014


  • quadratic algebras, Anick's conjecture, Golod-Shafarevich theorem

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