Skip to main content
SpringerLink
Log in

Twisted Heisenberg Doubles

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cautis S., Licata A.: Heisenberg categorification and Hilbert schemes. Duke Math. J. 161(13), 2469–2547 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. 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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Frenkel, I.B., Jing, N., Wang, W.: Vertex representations via finite groups and the McKay correspondence. Int. Math. Res. Not. (4), 195–222 (2000)

  4. 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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Khovanov, M.: Heisenberg Algebra and a Graphical Calculus. arXiv:1009.3295[math.RT]

  6. Licata A., Savage A.: Hecke algebras, finite general linear groups, and Heisenberg categorification. Quantum Topol. 4(2), 125–185 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lusztig, G.: Introduction to quantum groups. In: Modern Birkhäuser Classics. Birkhäuser/ Springer, New York (2010) (reprint of the 1994 edition)

  8. 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)

  9. 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)

  10. 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)

  11. Rosso, D., Savage, A.: Towers of Graded Superalgebras Categorify the Twisted Heisenberg Double. arXiv:1406.0421 [math.RT] (preprint)

  12. Savage A., Yacobi O.: Categorification and Heisenberg doubles arising from towers of algebras. J. Combin. Theory Ser. A 129, 19–56 (2015)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada

    Daniele Rosso & Alistair Savage

  2. Centre de Recherches Mathématiques, Montréal, Canada

    Daniele Rosso

Authors
  1. Daniele Rosso
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alistair Savage
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alistair Savage.

Additional information

Communicated by N. Reshetikhin

A. Savage was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. D. Rosso was supported by the Centre de Recherches Mathématiques and the Discovery Grant of A. Savage.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rosso, D., Savage, A. Twisted Heisenberg Doubles. Commun. Math. Phys. 337, 1053–1076 (2015). https://doi.org/10.1007/s00220-015-2330-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-015-2330-z

Keywords

Search

Navigation