Algebraic & Geometric Topology

Cohomology of Kac–Moody groups over a finite field

Jaume Aguadé and Albert Ruiz

Full-text: Open access

Abstract

We compute the mod p cohomology algebra of a family of infinite discrete Kac–Moody groups of rank two defined over finite fields of characteristic different from p.

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 4 (2013), 2207-2238.

Dates
Received: 26 June 2012
Revised: 12 March 2013
Accepted: 13 March 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715636

Digital Object Identifier
doi:10.2140/agt.2013.13.2207

Mathematical Reviews number (MathSciNet)
MR3073914

Zentralblatt MATH identifier
1301.55010

Subjects
Primary: 55R35: Classifying spaces of groups and $H$-spaces 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations [See also 17B65, 17B67, 22E65, 22E67, 22E70] 20G44: Kac-Moody groups

Keywords
cohomology classifying spaces Kac–Moody groups

Citation

Aguadé, Jaume; Ruiz, Albert. Cohomology of Kac–Moody groups over a finite field. Algebr. Geom. Topol. 13 (2013), no. 4, 2207--2238. doi:10.2140/agt.2013.13.2207. https://projecteuclid.org/euclid.agt/1513715636


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References

  • A Adem, R J Milgram, Cohomology of finite groups, 2nd edition, Grundl. Math. Wissen. 309, Springer, Berlin (2004)
  • J Aguadé, The arboreal approach to pairs of involutions in rank two, Comm. Algebra 37 (2009) 1104–1116
  • J Aguadé, $p$–compact groups as subgroups of maximal rank of Kac–Moody groups, J. Math. Kyoto Univ. 49 (2009) 83–112
  • J Aguadé, C Broto, N Kitchloo, L Saumell, Cohomology of classifying spaces of central quotients of rank two Kac–Moody groups, J. Math. Kyoto Univ. 45 (2005) 449–488
  • J Aguadé, C Broto, L Saumell, Rank two integral representations of the infinite dihedral group, Comm. Algebra 35 (2007) 1539–1551
  • J Aguadé, A Ruiz, Maps between classifying spaces of Kac–Moody groups, Adv. Math. 178 (2003) 66–98
  • C Broto, N Kitchloo, Classifying spaces of Kac–Moody groups, Math. Z. 240 (2002) 621–649
  • C Broto, J M Møller, Chevalley $p$–local finite groups, Algebr. Geom. Topol. 7 (2007) 1809–1919
  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1982)
  • A Dold, Lectures on algebraic topology, 2nd edition, Grundl. Math. Wissen. 200, Springer, Berlin (1980)
  • Z Fiedorowicz, S Priddy, Homology of classical groups over finite fields and their associated infinite loop spaces, Lecture Notes in Mathematics 674, Springer, Berlin (1978)
  • J D Foley, Comparing Kac–Moody groups over the complex numbers and fields of positive characteristic via homotopy theory, PhD thesis, University of California, San Diego (2012) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/1024147810 {\unhbox0
  • E M Friedlander, Computations of $K$–theories of finite fields, Topology 15 (1976) 87–109
  • A González, Unstable Adams operations acting on $p$–local compact groups and fixed points, Algebr. Geom. Topol. 12 (2012) 867–908
  • J Grodal, The classification of $p$–compact groups and homotopical group theory, from: “Proc. ICM Vol. II”, (R Bhatia, A Pal, G Rangarajan, V Srinivas, M Vanninathan, editors), Hindustan Book Agency, New Delhi (2010) 973–1001
  • V G Kac, Constructing groups associated to infinite-dimensional Lie algebras, from: “Infinite-dimensional groups with applications”, (V Kac, editor), Math. Sci. Res. Inst. Publ. 4, Springer, New York (1985) 167–216
  • V G Kac, D H Peterson, Defining relations of certain infinite-dimensional groups, from: “The mathematical heritage of Élie Cartan”, Astérisque, SMF, Paris (1985) 165–208
  • N Kitchloo, Topology of Kac–Moody groups, PhD thesis, MIT (1998) Available at \setbox0\makeatletter\@url http://www.math.jhu.edu/~nitu/papers/Thesis.pdf {\unhbox0
  • S A Mitchell, Quillen's theorem on buildings and the loops on a symmetric space, Enseign. Math. 34 (1988) 123–166
  • B Rémy, Groupes de Kac–Moody déployés et presque déployés, Astérisque 277, SMF, Paris (2002) viii+348
  • A Ruiz, Maps between classifying spaces of rank two Kac–Moody groups, PhD thesis, Universidad Autònoma de Barcelona (2001)
  • L Smith, Polynomial invariants of finite groups, Research Notes in Mathematics 6, A K Peters Ltd., Wellesley, MA (1995)
  • J Tits, Ensembles ordonnés, immeubles et sommes amalgamées, Bull. Soc. Math. Belg. Sér. A 38 (1986) 367–387
  • J Tits, Uniqueness and presentation of Kac–Moody groups over fields, J. Algebra 105 (1987) 542–573