Dual Lie Bialgebra Structures of W-algebra W(2, 2) Type

  • Guang Ai Song
  • Yu Cai Su
  • Xiao Qing YueEmail author


In the present paper, we investigate the dual Lie coalgebras of the centerless W(2, 2) algebra by studying the maximal good subspaces. Based on this, we construct the dual Lie bialgebra structures of the centerless W(2, 2) Lie bialgebra. As by-products, four new infinite dimensional Lie algebras are obtained.


W(2,2) algebra Lie bialgebra Lie coalgebra dual Lie bialgebra maximal good subspace 

MR(2010) Subject Classification

17B62 17B05 17B06 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Block, R.: Commutative Hopf algebras, Lie coalgebras, and divided powers. J. Algebra, 96, 275–306 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Block, R., Leroux, P.: Generalized dual coalgebras of algebras, with applications to cofree coalgebras. J. Pure Appl. Algebra, 36, 15–21 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Diarra, B.: On the definition of the dual Lie coalgebra of a Lie algebra. Publ. Mat., 39, 349–354 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Drinfel’d, V.: Hamlitonian structures on Lie group, Lie algebras and the geometric meaning of classical Yang—Baxter equations. Soviet Math. Dokl., 27, 68–71 (1983)zbMATHGoogle Scholar
  5. [5]
    Drinfel’d, V.: Quantum groups, Proceedings ICM (Berkeley 1986), Providence: Amer. Math. Soc., 789–820 1987Google Scholar
  6. [6]
    Griffing, G.: The dual coalgebra of certain infinite-dimensional Lie algebras. Comm. Algebra, 30, 5715–5724 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Li, J., Su, Y., Xin, B.: Lie bialgebra structures on the centerless W-algebra W(2, 2). Algebra Colloq., 17(2), 181–190 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Li, J., Su, Y.: Quantizations of the W-algebra W(2, 2). Acta Math. Sin., Engl. Ser., 27(4), 647–656 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Liu, D., Zhu, L.: Classification of Harish—Chandra over the W-algebra W(2, 2), preprint, arXiv:0801.2601v2Google Scholar
  10. [10]
    Liu, D., Gao, S., Zhu, L.: Classification of irreducible weight modules over W-algebra W(2, 2). J. Math. Phys., 49(11), 113503, 6 pp. (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Majid, S.: Foundations of Quantum Group Theory, Cambridge University Press, Cambridge 1995CrossRefzbMATHGoogle Scholar
  12. [12]
    Michaelis, W.: A class of infinite dimensional Lie bialgebras containing the Virasoro algebras. Adv. Math., 107, 365–392 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Michaelis, W.: The dual Lie bialgebra of a Lie bialgebra, Modular interfaces (Riverside, CA, 1995), 81–93, AMS/IP Stud. Adv. Math., 4, Amer. Math. Soc., Providence, RI, 1997Google Scholar
  14. [14]
    Nichols, W.: The structure of the dual Lie coalgebra of the Witt algebras. J. Pure Appl. Algebra, 68, 359–364 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Nichols, W.: On Lie and associative duals. J. Pure Appl. Algebra, 87, 313–320 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Peterson, B., Taft, E.: The Hopf algebra of linearly recursive sequences. Aequationes Mathematicae, 20, 1–17 University Waterloo (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Taft, E.: Witt and Virasoro algebras as Lie bialgebras. J. Pure Appl. Algebra, 87, 301–312 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Ng, S. H., Taft, E.: Classification of the Lie bialgebra structures on the Witt and Virasoro algebras. J. Pure Appl. Algebra, 151, 67–88 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Song, G., Su, Y.: Lie Bialgebras of generalized Witt type. Science in China A, 49, 533–544 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Song, G., Su, Y.: Dual Lie Bialgebras of Witt and Virasoro Types (in Chinese). Sci. Sin. Math., 43, 1093–1102 (2013)CrossRefGoogle Scholar
  21. [21]
    Song, G., Su, Y.: Dual Lie Bialgebra Structures of Poisson Types. Science in China A, 58, 1151–1162 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Song, G., Su, Y.: Dual Lie Bialgebra Structures of Loop and Current Virasooro Algebras (in Chinese). Sci. Sin. Math., 47, 1595–1606 (2017)CrossRefGoogle Scholar
  23. [23]
    Song, G., Su, Y., Yue, X.: Dual Lie bialgebra structures of the twisted Heisenberg—Virasoro type, arXiv:1703.03118v1 [math.RA] 9 Mar 2017Google Scholar
  24. [24]
    Su, Y., Zhao, K.: Generalized Virasoro and super-Virasoro algebras and modules of the intermediate series. J. Algebra, 252, 1–19 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Sweedler, M. E.: Hopf Algebras, New York: W A Benjamin Inc, 1969zbMATHGoogle Scholar
  26. [26]
    Wu, Y., Song, G., Su, Y.: Lie bialgebras of generalized Virasoro-like type. Acta Math. Sinica Engl. Ser., 22, 1915–1922 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Xin, B., Song, G., Su, Y.: Hamiltonian type Lie bialgebras. Science in China A, 50, 1267–1279 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Zhang, W., Dong, C.: W-algebra W(2, 2) and the Vertex operator algebra L(½, 0) ⊗ L(0, ½). Comm. Math. Phys., 285(3), 991–1004 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceShandong Technology and Business UniversityPenglai, ShandongP. R. China
  2. 2.School of Mathematical SciencesTongji UniversityShanghaiP. R. China

Personalised recommendations