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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. ). 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 [math.AG]).
Original language | English |
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Pages (from-to) | 1197-1212 |
Number of pages | 16 |
Journal | Selecta Mathematica (New Series) |
Volume | 20 |
Issue number | 4 |
Early online date | 17 Apr 2014 |
DOIs | |
Publication status | Published - Oct 2014 |
Keywords
- Golod-Shaferevich algebras
- Growth of algebras and the Gelfand-Kirillov dimension
- GELFAND-KIRILLOV DIMENSION
- POWER-SERIES RINGS
- BEZOUT
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Dive into the research topics of 'Growth, entropy and commutativity of algebras satisfying prescribed relations'. Together they form a unique fingerprint.Projects
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Agata Smoktunowicz
- School of Mathematics - Personal Chair in Algebra
Person: Academic: Research Active