Communications in Mathematical Physics

, Volume 337, Issue 3, pp 1053–1076 | Cite as

Twisted Heisenberg Doubles

  • Daniele Rosso
  • Alistair SavageEmail author


We introduce a twisted version of the Heisenberg double, constructed from a twisted Hopf algebra and a twisted pairing. We state a Stone–von Neumann type theorem for a natural Fock space representation of this twisted Heisenberg double and deduce the effect on the algebra of shifting the product and coproduct of the original twisted Hopf algebra. We conclude by showing that the quantum Weyl algebra, quantum Heisenberg algebras, and lattice Heisenberg algebras are all examples of the general construction.


Bilinear Form Hopf Algebra Dual Pair Monoidal Category Symmetric Bilinear Form 
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  1. CL12.
    Cautis S., Licata A.: Heisenberg categorification and Hilbert schemes. Duke Math. J. 161(13), 2469–2547 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  2. FJW00a.
    Frenkel I.B., Jing N., Wang W.: Quantum vertex representations via finite groups and the McKay correspondence. Commun. Math. Phys. 211(2), 365–393 (2000)CrossRefADSzbMATHMathSciNetGoogle Scholar
  3. FJW00b.
    Frenkel, I.B., Jing, N., Wang, W.: Vertex representations via finite groups and the McKay correspondence. Int. Math. Res. Not. (4), 195–222 (2000)Google Scholar
  4. FJW02.
    Frenkel I.B., Jing N., Wang W.: Twisted vertex representations via spin groups and the McKay correspondence. Duke Math. J. 111(1), 51–96 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  5. Kho.
    Khovanov, M.: Heisenberg Algebra and a Graphical Calculus. arXiv:1009.3295[math.RT]
  6. LS13.
    Licata A., Savage A.: Hecke algebras, finite general linear groups, and Heisenberg categorification. Quantum Topol. 4(2), 125–185 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. Lus10.
    Lusztig, G.: Introduction to quantum groups. In: Modern Birkhäuser Classics. Birkhäuser/ Springer, New York (2010) (reprint of the 1994 edition)Google Scholar
  8. LZ00.
    Li, L., Zhang, P.: Twisted Hopf algebras, Ringel–Hall algebras, and Green’s categories. J. Algebra 231(2), 713–743 (2000) (with an appendix by the referee)Google Scholar
  9. Mac95.
    Macdonald, I.G.: Symmetric functions and Hall polynomials. In: Oxford Mathematical Monographs, 2nd edn. The Clarendon Press, Oxford University Press, New York (1995) (with contributions by A. Zelevinsky, Oxford Science Publications)Google Scholar
  10. Rin96.
    Ringel, C.M.: Green’s theorem on Hall algebras. In: Representation Theory of Algebras and Related Topics (Mexico City, 1994). CMS Conference Proceedings, vol. 19, pp. 185–245. American Mathematical Society, Providence (1996)Google Scholar
  11. RS.
    Rosso, D., Savage, A.: Towers of Graded Superalgebras Categorify the Twisted Heisenberg Double. arXiv:1406.0421 [math.RT] (preprint)
  12. SY15.
    Savage A., Yacobi O.: Categorification and Heisenberg doubles arising from towers of algebras. J. Combin. Theory Ser. A 129, 19–56 (2015)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.Centre de Recherches MathématiquesMontréalCanada

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