Minimal Hilbert series for quadratic algebras and the Anick conjecture; pp. 301–305Full article in PDF format
| doi: 10.3176/proc.2010.4.08
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
– 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.
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.
3. Cameron, P. and Iyudu, N. Graphs of relations and Hilbert series. J. Symbolic Comput., 2007, 42, 1066–1078.
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/BF01247094Back to Issue