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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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Minimal Hilbert series for quadratic algebras and the Anick conjecture; pp. 301–305

(Full article in PDF format) doi: 10.3176/proc.2010.4.08


Authors

Natalia Iyudu, Stanislav Shkarin

Abstract

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  n2/4 and d  n2/2, 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(n2 + 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.

Keywords

quadratic algebras, Golod–Shafarevich theorem, Anick conjecture, Vershik conjecture.

References

  1. Anick , D. Generic algebras and CW complexes. In Algebraic Topology and Algebraic K-Theory. Princeton Univ. Press , N.J. , 1983 , 247–321.

  2. Anick , D. Noncommutative graded algebras and their Hilbert series. J. Algebra , 1982 , 78 , 120–140.
doi:10.1016/0021-8693(82)90104-1

  3. Cameron , P. and Iyudu , N. Graphs of relations and Hilbert series. J. Symbolic Comput. , 2007 , 42 , 1066–1078.
doi:10.1016/j.jsc.2007.07.006

  4. Golod , E. and Shafarevich , I. On the class field tower. Izv. Akad. Nauk SSSR Ser. Mat. , 1964 , 28 , 261–272 (in Russian).

  5. Golod , E. On nil algebras and residually finite p-groups. Izv. Akad. Nauk SSSR Ser. Mat. , 1964 , 28 , 273–276 (in Russian).

  6. Polishchuk , A. and Positselski , L. Quadratic Algebras. University Lecture Series 37. American Mathematical Society , Providence , RI , 2005.

  7. Ufnarovskii , V. Combinatorial and asymptotic methods in algebra. In Current Problems in Mathematics. Fundamental Directions , Vol. 57. Itogi Nauki i Tekhniki. Akad. Nauk SSSR , Moscow , 1990 , 5–177 (in Russian).

  8. Vershik , A. Algebras with quadratic relations. Selecta Math. Soviet , 1992 , 11 , 293–315.

  9. Zvyagina , M. Generic algebras with quadratic commutation relations. J. Soviet Math. , 1988 , 41 , 992–995.
doi:10.1007/BF01247094
 
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Current Issue: Vol. 68, Issue 4, 2019




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