Skip to main content
SpringerLink
Log in

Quiver Varieties and Tensor Products, II

  • Conference paper
Symmetries, Integrable Systems and Representations

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 40))

Abstract

We define a family of homomorphisms on a collection of convolution algebras associated with quiver varieties, which gives a kind of coproduct on the Yangian associated with a symmetric Kac-Moody Lie algebra. We study its property using perverse sheaves.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I, Luminy, 1981. Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982)

    Google Scholar 

  2. Bernstein, J., Lunts, V.: Equivariant Sheaves and Functors. Lecture Notes in Mathematics, vol. 1578. Springer, Berlin (1994)

    MATH  Google Scholar 

  3. Borho, W., MacPherson, R.: Partial resolutions of nilpotent varieties. In: Analysis and Topology on Singular Spaces, II, III, Luminy, 1981. Astérisque, vol. 101, pp. 23–74. Soc. Math. France, Paris (1983)

    Google Scholar 

  4. Braden, T.: Hyperbolic localization of intersection cohomology. Transform. Groups 8(3), 209–216 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birkhäuser, Boston (1997)

    MATH  Google Scholar 

  6. Crawley-Boevey, W.: Geometry of the moment map for representations of quivers. Compos. Math. 126(3), 257–293 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Drinfel’d, V.G.: Hopf algebras and the quantum Yang-Baxter equation. Dokl. Akad. Nauk SSSR 283(5), 1060–1064 (1985)

    MathSciNet  Google Scholar 

  8. Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Am. Math. Soc. 4(2), 365–421 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lusztig, G.: Introduction to Quantum Groups. Progress in Mathematics, vol. 110. Birkhäuser, Boston (1993)

    MATH  Google Scholar 

  10. Lusztig, G.: Cuspidal local systems and graded Hecke algebras. II. In: Representations of Groups, Banff, AB, 1994. CMS Conf. Proc., vol. 16, pp. 217–275. Amer. Math. Soc., Providence (1995). With errata for Part I [Inst. Hautes Études Sci. Publ. Math. No. 67, 145–202 (1988); MR0972345 (90e:22029)]

    Google Scholar 

  11. Malkin, A.: Tensor product varieties and crystals: the ADE case. Duke Math. J. 116(3), 477–524 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology. Preprint, arXiv:1211.1287

  13. Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76(2), 365–416 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nakajima, H.: Quiver varieties and Kac-Moody algebras. Duke Math. J. 91(3), 515–560 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nakajima, H.: Lectures on Hilbert Schemes of Points on Surfaces. University Lecture Series, vol. 18. American Mathematical Society, Providence (1999)

    MATH  Google Scholar 

  16. Nakajima, H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Am. Math. Soc. 14(1), 145–238 (2001) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nakajima, H.: Quiver varieties and tensor products. Invent. Math. 146(2), 399–449 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nakajima, H.: Geometric construction of representations of affine algebras. In: Proceedings of the International Congress of Mathematicians, vol. I, Beijing, 2002, pp. 423–438. Higher Ed. Press, Beijing (2002)

    Google Scholar 

  19. Nakajima, H.: Quiver varieties and branching. SIGMA 5, 003 (2009). 37 pages

    MathSciNet  Google Scholar 

  20. Nakajima, H.: AGT conjecture and convolution algebras (2012). http://www.kurims.kyoto-u.ac.jp/~nakajima/Talks/2012-04-04%20Nakajima.pdf

  21. Varagnolo, M., Vasserot, E.: Standard modules of quantum affine algebras. Duke Math. J. 111(3), 509–533 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Varagnolo, M., Vasserot, E.: Perverse sheaves and quantum Grothendieck rings. In: Studies in Memory of Issai Schur, Chevaleret/Rehovot, 2000. Progr. Math., vol. 210, pp. 345–365. Birkhäuser, Boston (2003)

    Chapter  Google Scholar 

Download references

Acknowledgements

The author thanks D. Maulik and A. Okounkov for discussion on their works. This work was supported by the Grant-in-aid for Scientific Research (No.23340005), JSPS, Japan.

Author information

Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Hiraku Nakajima

Authors
  1. Hiraku Nakajima
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiraku Nakajima .

Editor information

Editors and Affiliations

  1. Inst. Camille Jordan, UMR5028 CNRS, Université Claude Bernard Lyon 1, Villeurbanne Cedex, 69622, France

    Kenji Iohara

  2. Inst. de Mathém. de Jus., UMR 7586 CNRS, Université Pierre et Marie Curie, Place Jussieu, Case 247 4, Paris, 75252, France

    Sophie Morier-Genoud

  3. Inst. Camille Jordan, UMR5028 CNRS, Université Claude Bernard Lyon I, Villeurbanne Cedex, 69622, France

    Bertrand Rémy

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag London

About this paper

Cite this paper

Nakajima, H. (2013). Quiver Varieties and Tensor Products, II. In: Iohara, K., Morier-Genoud, S., Rémy, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/978-1-4471-4863-0_16

Download citation

Publish with us

Policies and ethics

Search

Navigation