Optimal 5-step nilpotent quadratic algebras

Stanislav Shkarin; Natalia Iyudu

doi:10.1016/j.jalgebra.2014.04.015

Optimal 5-step nilpotent quadratic algebras

Stanislav Shkarin, Natalia Iyudu

Research output: Contribution to journal › Article › peer-review

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Abstract

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 language	English
Pages (from-to)	1-14
Number of pages	14
Journal	Journal of Algebra
Volume	412
Early online date	17 May 2014
DOIs	https://doi.org/10.1016/j.jalgebra.2014.04.015
Publication status	Published - 15 Aug 2014

Keywords

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

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10.1016/j.jalgebra.2014.04.015Licence: Unspecified

Stanislav Shkarin
- s.shkarinqub.acuk
- School of Mathematics and Physics - Lecturer
- Mathematical Sciences Research Centre
Person: Academic

Cite this

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title = "Optimal 5-step nilpotent quadratic algebras",

abstract = "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.",

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author = "Stanislav Shkarin and Natalia Iyudu",

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AU - Iyudu, Natalia

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AB - 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.

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U2 - 10.1016/j.jalgebra.2014.04.015

DO - 10.1016/j.jalgebra.2014.04.015

M3 - Article

SN - 0021-8693

VL - 412

SP - 1

EP - 14

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Optimal 5-step nilpotent quadratic algebras

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Stanislav Shkarin

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