Minimal Hilbert Series for Quadratic Algebras and the Anick Conjecture

Ruutalgebrate minimaalsed Hilberti read ja Anicki hüpotees

Natalia Iyudu ； Stanislav Shkarin

《Proceedings of the Estonian Academy of Sciences》 59卷4期 (2010/12) Pp. 301-305

並列摘要

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≤n^2/4 and d≥n^2/2, and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to n(n-1)/2 was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional.
We announce here the result that over any infinite field, the Anick conjecture holds for d≥4(n^2+n)/9 and an arbitrary number of generators. We also discuss the result that confirms the Vershik conjecture over any field of characteristic 0, and a series of related asymptotic results.

並列關鍵字

quadratic algebras ； Golod-Shafarevich theorem ； Anick conjecture ； Vershik conjecture