The Michigan Mathematical Journal

On the homotopy Lie algebra of an arrangement

Graham Denham and Alexander I. Suciu

Full-text: Open access

Article information

Source
Michigan Math. J. Volume 54, Issue 2 (2006), 319-340.

Dates
First available in Project Euclid: 23 August 2006

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1156345597

Digital Object Identifier
doi:10.1307/mmj/1156345597

Mathematical Reviews number (MathSciNet)
MR2252762

Zentralblatt MATH identifier
1198.17012

Subjects
Primary: 16E05: Syzygies, resolutions, complexes 52C35: Arrangements of points, flats, hyperplanes [See also 32S22]
Secondary: 16S37: Quadratic and Koszul algebras 55P62: Rational homotopy theory

Citation

Denham, Graham; Suciu, Alexander I. On the homotopy Lie algebra of an arrangement. Michigan Math. J. 54 (2006), no. 2, 319--340. doi:10.1307/mmj/1156345597. https://projecteuclid.org/euclid.mmj/1156345597


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References

  • L. Avramov, Small homomorphisms of local rings, J. Algebra 50 (1978), 400--453.
  • ------, Golod homomorphisms, Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture Notes in Math., 1183, pp. 59--78, Springer-Verlag, Berlin, 1986.
  • ------, Infinite free resolutions, Six lectures on commutative algebra (Bellaterra, 1996), Progr. Math., 166, pp. 1--118, Birkhäuser, Basel, 1998.
  • A. Björner and G. Ziegler, Broken circuit complexes: Factorizations and generalizations, J. Combin. Theory Ser. B 51 (1991), 96--126.
  • D. C. Cohen, F. R. Cohen, and M. Xicoténcatl, Lie algebras associated to fiber-type arrangements, Internat. Math. Res. Notices 29 (2003), 1591--1621.
  • G. Denham and S. Yuzvinsky, Annihilators of Orlik--Solomon relations, Adv. in Appl. Math. 28 (2002), 231--249.
  • A. Dimca and S. Papadima, Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments, Ann. of Math. (2) 158 (2003), 473--507.
  • D. Eisenbud, S. Popescu, and S. Yuzvinsky, Hyperplane arrangement cohomology and monomials in the exterior algebra, Trans. Amer. Math. Soc. 355 (2003), 4365--4383.
  • M. Falk, The minimal model of the complement of an arrangement of hyperplanes, Trans. Amer. Math. Soc. 309 (1988), 543--556.
  • ------, Line-closed matroids, quadratic algebras, and formal arrangements, Adv. in Appl. Math. 28 (2002), 250--271.
  • M. Falk and R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), 77--88.
  • Y. Félix and J.-C. Thomas, On the ubiquity of the rational homotopy Lie algebra of a topological space, Bull. Soc. Math. Belg. Sér. A 38 (1986), 175--190.
  • R. Fröberg, Koszul algebras, Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Appl. Math., 205, pp. 337--350, Dekker, New York, 1999.
  • R. Fröberg and C. Löfwall, Koszul homology and Lie algebras with application to generic forms and points, Homology Homotopy Appl. 4 (2002), 227--258.
  • D. Grayson and M. Stillman, Macaulay 2: A software system for research in algebraic geometry; available at $\langle $http://www.math.uiuc.edu/Macaulay2$\rangle .$
  • M. Jambu and S. Papadima, A generalization of fiber-type arrangements and a new deformation method, Topology 37 (1998), 1135--1164.
  • ------, Deformations of hypersolvable arrangements, Arrangements in Boston: A conference on hyperplane arrangements (1999), Topology Appl. 118 (2002), 103--111.
  • P. Jørgensen, Non-commutative Castelnuovo--Mumford regularity, Math. Proc. Cambridge Philos. Soc. 125 (1999), 203--221.
  • C. Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra, Algebra, algebraic topology and their interactions (Stockholm, 1983), Lecture Notes in Math., 1183, pp. 291--338, Springer-Verlag, Berlin, 1986.
  • J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211--264.
  • H. K. Nandi, Enumeration of non-isomorphic solutations of balanced incomplete block designs, Sankhy\B a 7 (1946), 305--312.
  • P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren Math. Wiss., 300, Springer-Verlag, Berlin, 1992.
  • S. Papadima and A. I. Suciu, Higher homotopy groups of complements of complex hyperplane arrangements, Adv. Math. 165 (2002), 71--100.
  • ------, Homotopy Lie algebras, lower central series, and the Koszul property, Geom. Topol. 8 (2004), 1079--1125.
  • S. Papadima and S. Yuzvinsky, On rational $K[\pi ,1]$ spaces and Koszul algebras, J. Pure Appl. Algebra 144 (1999), 157--167.
  • D. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411--418.
  • V. V. Schechtman and A. N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139--194.
  • H. K. Schenck and A. I. Suciu, Resonance, linear syzygies, Chen groups, and the Bernstein--Gelfand--Gelfand correspondence, Trans. Amer. Math. Soc. 358 (2006), 2269--2289.
  • D. W. Sharpe and P. Vámos, Injective modules, Cambridge Tracts Math. Math. Phys., 62, Cambridge Univ. Press, London, 1972.
  • B. Shelton and S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. (2) 56 (1997), 477--490.
  • D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269--331.
  • S. Yuzvinsky, Small rational model of subspace complement, Trans. Amer. Math. Soc. 354 (2002), 1921--1945.