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Growth, entropy and commutativity of algebras satisfying prescribed relations

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Abstract

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfies some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod–Shafarevich algebras. This paper provides bounds for the growth function on images of Golod–Shafarevich algebras based upon the number of defining relations. This extends results from Smoktunowicz and Bartholdi (Q J Math. doi:10.1093/qmath/hat005 2013) and Smoktunowicz (J Algebra 381:116–130, 2013). Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky (A private communication, 2013) by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov (2013). Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss (Noncommutative deformations and flops, ArXiv:1309.0698v2 [math.AG]).

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Acknowledgments

This research was funded by ERC Advanced Grant 320974. The author is very grateful to Ivan Chelstov for providing the references for Lemma 5.1, to Efim Zelmanov for his inspiring questions, and to Michael West for his continued assistance with my written English.

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  1. Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, JCMB, King’s Buildings, Mayfield Road, Edinburgh , EH9 3JZ, Scotland, UK

    Agata Smoktunowicz

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Smoktunowicz, A. Growth, entropy and commutativity of algebras satisfying prescribed relations. Sel. Math. New Ser. 20, 1197–1212 (2014). https://doi.org/10.1007/s00029-014-0154-x

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