Graph products of spheres, associative graded algebras and Hilbert series. (English) Zbl 1273.16022
Summary: Given a finite, simple, vertex-weighted graph, we construct a graded associative (noncommutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it is easy to recognize if the ideal is ‘inert’, from which strong results on the algebra follow. Noncommutative Gröbner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment-angle complex. We apply our result to the loop-space homology of this space.
MSC:
| 16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 05C31 | Graph polynomials |
| 55P35 | Loop spaces |
| 16W50 | Graded rings and modules (associative rings and algebras) |
Keywords:
graded algebras; finite simple vertex-weighted graphs; generators; ideals of relations; partial products of spheres; Hilbert series; Gröbner bases; toric topology; loop-space homologySoftware:
BERGMANReferences:
| [1] | Adams J.F., Hilton P.J.: On the chain algebra of a loop space. Comment. Math. Helv. 30, 305–330 (1956) · Zbl 0071.16403 · doi:10.1007/BF02564350 |
| [2] | Anick D.J.: A counterexample to a conjecture of Serre. Ann. Math. (2) 115(1), 1–33 (1982) · doi:10.2307/1971338 |
| [3] | Anick D.J.: Noncommutative graded algebras and their Hilbert series. J. Algebra 78(1), 120–140 (1982) · Zbl 0502.16002 · doi:10.1016/0021-8693(82)90104-1 |
| [4] | Anick, D.J.: Connections between Yoneda and Pontrjagin algebras. In: Algebraic Topology, Aarhus 1982 (Aarhus, 1982). Lecture Notes in Mathematics, vol. 1051, pp. 331–350. Springer, Berlin (1984) |
| [5] | Backelin, J.: BERGMAN. Available at http://servus.matematik.su.se/bergman |
| [6] | Bahri, A., Bendersky, M., Cohen, F.R., Gitler, S.: The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces. arXiv:0711.4689 [math.AT] (2007) · Zbl 1197.13021 |
| [7] | Bergman G.: The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978) · doi:10.1016/0001-8708(78)90010-5 |
| [8] | Bubenik P.: Free and semi-inert cell attachments. Trans. Am. Math. Soc. 357(11), 4533–4553 (2005) · Zbl 1073.55007 · doi:10.1090/S0002-9947-05-03989-9 |
| [9] | Bubenik P.: Separated Lie models and the homotopy Lie algebra. J. Pure Appl. Algebra 212(2), 401–410 (2008) · Zbl 1128.55009 · doi:10.1016/j.jpaa.2007.05.018 |
| [10] | Bukhshtaber V.M., Panov T.E.: Actions of tori, combinatorial topology and homological algebra. Russ. Math. Surv. 55(5), 825–921 (2000) · Zbl 1010.52011 · doi:10.1070/RM2000v055n05ABEH000320 |
| [11] | Cartier P., Foata D.: Problèmes combinatoires de commutation et réarrangements. In: Lecture Notes in Mathematics, vol. 85. Springer, Berlin (1969) · Zbl 0186.30101 |
| [12] | Davis M.W., Januszkiewicz T.: Convex polytopes, coxeter orbifolds and torus actions. Duke Math. J. 62(2), 417–451 (1991) · Zbl 0733.52006 · doi:10.1215/S0012-7094-91-06217-4 |
| [13] | Denham G., Suciu A.I.: Moment-angle complexes, monomial ideals and Massey products. Pure Appl. Math. Q. 3(1, part 3), 25–60 (2007) · Zbl 1169.13013 · doi:10.4310/PAMQ.2007.v3.n1.a2 |
| [14] | Diekert, V., Rozenberg, G. (eds): The Book of Traces. World Scientific Publishing, River Edge (1995) |
| [15] | Diekert, V., Métivier, Y.: Partial commutation and traces. In: Handbook of Formal Languages, vol. 3, pp. 457–533. Springer, Berlin (1997) |
| [16] | Dobrinskaya, N.: Loops on polyhedral products and diagonal arrangements. arXiv:0901.2871 [math.AT] (2009) · Zbl 1197.55004 |
| [17] | Félix Y., Halperin S., Thomas J.-C.: Rational homotopy theory. In: Graduate Texts in Mathematics, vol. 205. Springer, New York (2001) · Zbl 0961.55002 |
| [18] | Félix Y., Tanré D.: Rational homotopy of the polyhedral product functor. Proc. Am. Math. Soc. 137(3), 891–898 (2009) · Zbl 1160.55008 · doi:10.1090/S0002-9939-08-09591-9 |
| [19] | Félix Y., Thomas J.-C.: Effet d’un attachement cellulaire dans l’homologie de l’espace des lacets. Ann. Inst. Fourier (Grenoble) 39(1), 207–224 (1989) · Zbl 0686.55005 · doi:10.5802/aif.1164 |
| [20] | Fröberg, R.: Koszul algebras. In: Advances in commutative ring theory (Fez, 1997). In: Lecture Notes in Pure and Applied Mathematics, vol. 205, pp. 337–350. Dekker, New York (1999) · Zbl 0962.13009 |
| [21] | Fröberg R.: Determination of a class of Poincaré series. Math. Scand. 37(1), 29–39 (1975) · Zbl 0318.13027 |
| [22] | Grbić J., Theriault S.: The homotopy type of the complement of a coordinate subspace arrangement. Topology 46(4), 357–396 (2007) · Zbl 1118.55006 · doi:10.1016/j.top.2007.02.006 |
| [23] | Halperin S., Lemaire J.-M.: Suites inertes dans les algèbres de Lie graduées (”Autopsie d’un meurtre. II”). Math. Scand. 61(1), 39–67 (1987) · Zbl 0655.55004 |
| [24] | Kim K.H., Roush F.W.: Homology of certain algebras defined by graphs. J. Pure Appl. Algebra 17, 179–186 (1980) · Zbl 0444.05066 · doi:10.1016/0022-4049(80)90083-3 |
| [25] | Kim K.H., Makar-Limanov L., Neggers J., Roush F.W.: Graph algebras. J. Algebra 64(1), 46–51 (1980) · Zbl 0431.05023 · doi:10.1016/0021-8693(80)90131-3 |
| [26] | Lallement G.: Semigroups and Combinatorial Applications. Wiley, New York (1979) · Zbl 0421.20025 |
| [27] | Lemaire Jean-Michel: Autopsie d’un meurtre” dans l’homologie d’une algèbre de chaînes. Ann. Sci. École Norm. Sup. (4) 11(1), 93–100 (1978) · Zbl 0382.55011 |
| [28] | Mazurkiewicz, A.: Concurrent Program Schemes and their Interpretations. Technical Report DAIMI PB 78, Aarhus University, Aarhus (1977) |
| [29] | Mikhalev, A.A., Zolotykh, A.A.: Combinatorial Aspects of Lie Superalgebras. CRC Press, Boca Raton [with 1 IBM-PC floppy disk (3.5 inch; HD)] (1995) |
| [30] | Mora, F.: Groebner bases for noncommutative polynomial rings. In: Algebraic Algorithms and Error Correcting Codes (Grenoble, 1985). Lecture Notes in Computer Science, vol. 229, pp. 353–362. Springer, Berlin (1986) |
| [31] | Notbohm D., Ray N.: On Davis–Januszkiewicz homotopy types. I. Formality and rationalisation. Algebraic Geom. Topol. 5, 31–51 (2005) (electronic) · Zbl 1065.55006 · doi:10.2140/agt.2005.5.31 |
| [32] | Panov, T. E., Ray, N.: Categorical aspects of toric topology. In: Toric Topology. Contemporary Mathematics, vol. 460, pp. 293–322. American Mathematical Society, Providence (2008) · Zbl 1149.55014 |
| [33] | Papadima S., Suciu A.I.: Algebraic invariants for right-angled Artin groups. Math. Ann. 334(3), 533–555 (2006) · Zbl 1165.20032 · doi:10.1007/s00208-005-0704-9 |
| [34] | Selick, P.: Introduction to homotopy theory. In: Fields Institute Monographs, vol. 9. American Mathematical Society, Providence (1997) · Zbl 0883.55001 |
| [35] | Shelton B., Yuzvinsky S.: Koszul algebras from graphs and hyperplane arrangements. J. Lond. Math. Soc. (2) 56(3), 477–490 (1997) · Zbl 0959.16026 · doi:10.1112/S0024610797005553 |
| [36] | Strickland, N.P.: Toric spaces. http://neil-strickland.staff.shef.ac.uk/papers/ (1999) |
| [37] | Tsejtin G.S.: Associative computations with unsolvable equivalence problem. Tr. Mat. Inst. Steklova 52, 172–189 (1958) |
| [38] | Ufnarovski, V.: Introduction to noncommutative Gröbner bases theory. In: Gröbner Bases and Applications (Linz, 1998). London Mathematical Society Lecture Note Series, vol. 251, pp. 259–280. Cambridge University Press, Cambridge (1998) · Zbl 0902.16002 |
| [39] | Whitehead J.H.C.: On adding relations to homotopy groups. Ann. Math. (2) 42, 409–428 (1941) · Zbl 0027.26404 · doi:10.2307/1968907 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
