Skip to main content
SpringerLink
Log in
Book cover

Affine, Vertex and W-algebras pp 99–134Cite as

Classical and Quantum \({\mathcal {W}}\)-Algebras and Applications to Hamiltonian Equations

  • Chapter
  • First Online:

Part of the book series: Springer INdAM Series ((SINDAMS,volume 37))

Abstract

We start by giving an overview of the four fundamental physical theories, namely classical mechanics, quantum mechanics, classical field theory and quantum field theory, and the corresponding algebraic structures, namely Poisson algebras, associative algebras, Poisson vertex algebras and vertex algebras. We then focus on classical and quantum \(\mathcal {W}\)-algebras, with a particular emphasis on their application to integrable Hamiltonian PDE.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
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

Learn about institutional subscriptions

References

  1. Adler, M.: On a trace functional for formal pseudodifferential operators and the symplectic structure of the Korteweg-de Vries equation. Invent. Math. 50, 219–248 (1979)

    Article  MATH  Google Scholar 

  2. Bakalov, B., Kac, V.G.: Field Algebras. IMRN 3, 123–159 (2003)

    Google Scholar 

  3. Bakalov, B., De Sole, A., Heluani, R., Kac, V.G.: An operadic approach to vertex algebra and Poisson vertex algebra cohomology. arXiv:1806.08754

  4. Bakalov, B., De Sole, A., Heluani, R., Kac, V.G.: Chiral vs classical operad. arXiv:1812.05972

  5. Bakalov, B., De Sole, A., Heluani, R., Kac, V.G.: Computation of cohomology of Lie conformal and Poisson vertex algebras. arXiv:19xxx

  6. Barakat, A., De Sole, A., Kac, V.G.: Poisson vertex algebras in the theory of Hamiltonian equations. Jpn. J. Math. 4(2), 141–252 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Borcherds, R.: Vertex algebras, Kac-Moody algebras and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Burroughs, N., de Groot, M., Hollowood, T., Miramontes, L.: Generalized Drinfeld-Sokolov hierarchies II: the Hamiltonian structures. Commun. Math. Phys. 153, 187–215 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. de Boer, J., Tjin, T.: Representation theory of finite W algebras. Commun. Math. Phys. 158, 485–516 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. de Boer, J., Tjin, T.: The relation between quantum W algebras and Lie algebras. Commun. Math. Phys. 160, 317–332 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. de Groot, M., Hollowood, T., Miramontes, L.: Generalized Drinfeld-Sokolov hierarchies. Commun. Math. Phys. 145, 57–84 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. De Sole, A., Kac, V.G.: Finite vs. affine W-algebras. Jpn. J. Math. 1(1), 137–261 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. De Sole, A.: On classical finite and affine W-algebras. Springer INdAM Ser. 7, 51–67 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. De Sole, A., Fedele, L., valeri, D.: Generators of the quantum finite W-algebras in type A. arXiv:1806.03233

  15. De Sole, A., Kac, V.G.: The variational Poisson cohomology Japan. J. Math. 8, 1–145 (2013)

    MATH  Google Scholar 

  16. De Sole, A., Kac, V.G.: Essential variational Poisson cohomology Comm. Math. Phys. 313(3), 837–864 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. De Sole, A., Kac, V.G.: Non-local Poisson structures and applications to the theory of integrable systems Jpn. J. Math. 8(2), 233–347 (2013)

    MathSciNet  MATH  Google Scholar 

  18. De Sole, A., Kac, V.G., Valeri, D.: Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras. Commun. Math. Phys. 323(2), 663–711 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. De Sole, A., Kac, V.G., Valeri, D.: Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents. Alberto De Sole, Victor G. Kac, Daniele Valeri Commun. Math. Phys. 331(2), 623–676 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. De Sole, A., Kac, V.G., Valeri, D.: Dirac reduction for Poisson vertex algebras Comm. Math. Phys. 331(3), 1155–1190 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. De Sole, A., Kac, V.G., Valeri, D.: Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras. Int. Math. Res. Not. 21, 11186–11235 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. De Sole, A., Kac, V.G., Valeri, D.: Integrability of Dirac reduced bi-Hamiltonian equations. Trends in Contemporary Mathematics. Springer INdAM Series vol. 8, pp. 13–32 (2014)

    Google Scholar 

  23. De Sole, A., Kac, V.G., Valeri, D.: Structure of classical (finite and affine) W-algebras. J. Eur. Math. Soc. 18(9), 1873–1908 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. De Sole, A., Kac, V.G., Valeri, D.: Double Poisson vertex algebras and non-commutative Hamiltonian equations. Adv. Math. 281, 1025–1099 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. De Sole, A., Kac, V.G., Valeri, D.: A new scheme of integrability for (bi)Hamiltonian PDE. Commun. Math. Phys. 347(2), 449–488 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. De Sole, A., Kac, V.G., Valeri, D.: Classical W-algebras for \(\mathfrak{gl}_N\) and associated integrable Hamiltonian hierarchies. Commun. Math. Phys. 348(1), 265–319 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. De Sole, A., Kac, V.G., Valeri, D.: Finite W-algebras for \(\mathfrak{gl}_N\). Adv. Math. 327, 173–224 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. De Sole, A., Kac, V.G., Valeri, D.: Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras. Commun. Math. Phys. 360(3), 851–918 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. De Sole, A., Kac, V.G., Valeri, D.: A Lax type operator for quantum finite W-algebras. arXiv:1707.03669

  30. De Sole, A., Kac, V.G., Valeri, D., Wakimoto, M.: Local and non-local multiplicative Poisson vertex algebras and differential-difference equations. arXiv:1809.01735

  31. De Sole, A., Kac, V.G., Valeri, D., Wakimoto, M.: Poisson \(\lambda \)-brackets for differential-difference equations. arXiv:1806.05536

  32. Drinfeld, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type. Sov. J. Math. 30, 1975–2036 (1985)

    Article  MATH  Google Scholar 

  33. Elashvili, A.G., Kac, V.G., Vinberg, E.B.: Cyclic elements in semisimple Lie algebras. Transf. Groups 18, 97–130 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fehér, L., Harnad, J., Marshall, I.: Generalized Drinfeld-Sokolov reductions and KdV type hierarchies. Commun. Math. Phys. 154(1), 181–214 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  35. Feigin, B.L., Frenkel, E.: Quantization of Drinfeld-Sokolov reduction. Phys. Lett., B 246, 75–81 (1990)

    Google Scholar 

  36. Fernández-Pousa, C., Gallas, M., Miramontes, L., Sánchez Guillén, J.: \(\cal{W}\)-algebras from soliton equations and Heisenberg subalgebras. Ann. Phys. 243(2), 372–419 (1995)

    Google Scholar 

  37. Fernández-Pousa, C., Gallas, M., Miramontes, L., Sánchez Guillén, J.: Integrable systems and \(\cal{W}\)-algebras. VIII J. A. Swieca Summer School on Particles and Fields (Rio de Janeiro, 1995), pp. 475–479

    Google Scholar 

  38. Gan, W.L., Ginzburg, V.: Quantization of Slodowy slices. Int. Math. Res. Not. 5, 243–255 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gelfand, I.M., Gelfand, S.I., Retakh, V., Wilson, R.L.: Quasideterminants. Adv. Math. 193(1), 56–141 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kac, V.G.: Vertex Algebras for Beginners. University Lecture Notes, vol. 10, 2nd edn. AMS, Providence (1996)

    MATH  Google Scholar 

  41. Kac, V.G., Roan, S.-S., Wakimoto, M.: Quantum reduction for affine superalgebras. Commun. Math. Phys. 241, 307–342 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kac, V.G., Wakimoto, M.: Quantum reduction and representation theory of superconformal algebras. Adv. Math. 185, 400–458 (2004). Corrigendum, Adv. Math. 193, 453–455 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kostant, B.: On Whittaker vectors and representation theory. Invent. Math. 48, 101–184 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  44. Lynch, T.E.: Generalized Whittaker vectors and representation theory. Thesis (Ph.D.) - Massachusetts Institute of Technology (1979)

    Google Scholar 

  45. Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19(5), 1156–1162 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  46. Premet, A.: Special transverse slices and their enveloping algebras. Adv. Math. 170, 1–55 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. Slodowy, P.: Simple Singularities and Simple Algebraic Groups. Lecture Notes in Mathematics, vol. 815. Springer, Berlin (1980)

    Book  MATH  Google Scholar 

  48. Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Progress in Mathematics, vol. 118. Birkhäuser, Basel (1994)

    Book  MATH  Google Scholar 

  49. Zamolodchikov, A.: Infinite extra symmetries in two-dimensional conformal quantum field theory. Teor. Mat. Fiz 65(3), 347–359 (1985)

    Article  MathSciNet  Google Scholar 

  50. Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. AMS 9, 237–302 (1996)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Rome La Sapienza, Roma, Italy

    Alberto De Sole

Authors
  1. Alberto De Sole
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto De Sole .

Editor information

Editors and Affiliations

  1. Faculty of Science and Department of Mathematics, University of Zagreb, Zagreb, Croatia

    Dražen Adamović

  2. Department of Mathematics, Sapienza University of Rome, Rome, Italy

    Paolo Papi

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

De Sole, A. (2019). Classical and Quantum \({\mathcal {W}}\)-Algebras and Applications to Hamiltonian Equations. In: Adamović, D., Papi, P. (eds) Affine, Vertex and W-algebras. Springer INdAM Series, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-030-32906-8_6

Download citation

Publish with us

Policies and ethics

Search

Navigation