arXiv 2201.044792022-01-12math.RAmath-phmath.AGmath.GRmath.MP

Golod-Shafarevich-Vinberg type theorems and finiteness conditions for potential algebras

Natalia Iyudu, Stanislav Shkarin

备注：: arXiv admin note: substantial text overlap with arXiv:1806.06829

英文摘要：

We obtain a lower estimate for the Hilbert series of Jacobi algebras and their completions by providing analogue of the Golog-Shafarevich-Vinberg theorem for potential case. We especially treat non-homogeneous situation. This estimate allows to answer number of questions arising in the work of Wemyss-Donovan-Brown on noncommutative singularities and deformation theory. In particular, we prove that the only case when a potential algebra or its completion could be finite dimensional or of linear growth, is the case of two variables and potential having terms of degree three.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A remark on Golod--Shafarevich algebras

We show that a direct limit of surjections of (weak) Golod--Shafarevich algebras is a weak Golod--Shafarevich algebra as well. This holds both for graded and for filtered algebras provided that the filtrations are induced by the filtration of the first entry of the sequence. It follows that the limit is an algebra of exponential growth. An example shows that the assumptions of this theorem cannot be directly weakened.

2014-12-30math.GRmath.RA

A mnemonic for the graded-case Golod-Shafarevich inequality

We draw attention to an easy-to-remember explanation for the graded-case inequality of Golod and Shafarevich. We review some of the classic material on this inequality.

2017-04-18math.RA

Golod-Shafarevich algebras, free subalgebras and Noetherian images

It is shown that Golod-Shaferevich algebras of a reduced number of defining relations contain noncommutative free subalgebras in two generators, and that these algebras can be homomorphically mapped onto prime, Noetherian algebras with linear growth. It is also shown that Golod-Shafarevich algebras of a reduced number of relations cannot be nil.

2012-12-18math.RA

The Golod Shafarevich counter-example without Hilbert series

Let $F$ be an arbitrary field. The Golod-Shafarevich example of a finitely generated nil $F$-algebra which is infinite dimensional -- is revisited. Here we offer a rather elementary treatment of that example, in which induction replaces Hilbert series techniques. This note also contains a detailed exposition of the construction of that example.

2009-01-11math.RA

Images of Golod-Shafarevich algebras with small growth

We show that Golod-Shafarevich algebras can be homomorphically mapped onto infinite-dimensional algebras with polynomial growth, under mild assumptions of the number of relations of given degrees. In case these algebras are finitely presented, we show they can be mapped onto an infinite dimensional algebras with quadratic growth. This answers a guestion by Zelmanov. We then show, by an elementary construction, that any sufficiently regular function at least $n^{\log n}$ may occur as the growth of an algebra.

2012-09-13math.RA

Golod-Shafarevich type theorems and potential algebras

Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than $8$. This answers a question of Wemyss \c…

2017-11-21math-phmath.GRmath.MP

更多相关论文收起更多论文

The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

We study the question on whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper 'Generic algebras and CW-complexes', Princeton Univ. Press., where he proved that the estimate is attained for the number of quadratic relations $d \leq \frac{n^2}{4}$ and $d \geq \frac{n^2}{2}$, and conjectured that this is the case for any number of quadratic relations. The particular point where the number of relations is equal to $ \frac{n(n-1)}{2}$ was addressed by Vershi…

2010-07-22math.QAmath.RA

Growth, entropy and commutativity of algebras satisfying prescribed relations

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfy 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 [32], [33]. 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 a…

2014-03-31math.RA

Asymptotically optimal $k$-step nilpotency of quadratic algebras and the Fibonacci numbers

It follows from the Golod--Shafarevich theorem that if R is an associative algebra given by n generators and $d<\frac{n^2}{4}\cos^{-2}(\fracπ{k+1})$ quadratic relations, then R is not k-step nilpotent. We show that the above estimate is asymptotically optimal, and establish number of related results. For example, we show that for any k this estimate is attained for ifinitely many n.

2016-01-04math.RA

Growth and Relations in Graded Rings

Suppose $A$ is a graded associative algebra over a field, $I$ is its ideal generated by a set $α$ of homogeneous elements, and B = A/I. In this note, some inequalities between Hilbert series of algebras $A,B$ and the number of elements of the set $α$ are announced. As in the Golod--Shafarevich inequality as in our case the equality in every estimate is exact iff the set $α$ is strongly free: so we obtain some new characterizations of such sets. As a consequence it is proved that over a field of zero characteristic for the class of finitely defined graded algebras there is no algorithm to answe…

1999-03-05math.RA

Three dimensional Sklyanin algebras and Groebner bases

We consider a Sklyanin algebra S with 3 generators, which is the quadratic algebra over a field k with three generators x,y,z given by three relations pxy+qyx+rzz=0, pyz+qzy+rxx=0 and pzx+qxz+ryy=0, where p,q,r are parameters from the fileld k. This class of algebras enjoyed much of attention, in particular, using tools from algebraic geometry, Feigin & Odesskii, and Artin, Tate & Van den Berg, showed that if at least two of the parameters p, q and r are non-zero and at least two of three numbers p^3,q^3 and r^3 are distinct, then S is Koszul and has the same Hilbert series as the algebra of c…

2016-08-27math-phmath.MPmath.RA