Geometriae Dedicata

, Volume 122, Issue 1, pp 145–157 | Cite as

Classical r-Matrices and Novikov Algebras

  • Dietrich Burde
Original Article


We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present a 2-step solvable Lie algebra without any Novikov structure. We use extensions and classical r-matrices to construct Novikov structures on certain classes of solvable Lie algebras.


r-matrices Novikov algebras Affine structures 

Mathematics Subject Classifications (2000)

Primary 17B30 17D25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balinskii A.A., Novikov S.P. (1985). Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras. Soviet Math. Dokl. 32:228–231Google Scholar
  2. 2.
    Benoist Y. (1995). Une nilvariété non affine. J. Differential Geom. 41:21–52MATHMathSciNetGoogle Scholar
  3. 3.
    Burde D. (1996). Affine structures on nilmanifolds. Int. J. Math. 7:599–616MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Burde, D., Dekimpe, K.: Novikov structures on solvable Lie algebras. J. Geom. Phys. (2006) (to appear)Google Scholar
  5. 5.
    Dorfman I., Gel’fand I.M. (1979). Hamiltonian operators and algebraic structures associated with them. Funktsional. Anal. i Prilozhen. 13:13–30MATHMathSciNetGoogle Scholar
  6. 6.
    Frenkel I., Huang Y.Z., Lepowsky L. (1993). On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 494:1–64MathSciNetGoogle Scholar
  7. 7.
    Gel’fand I.M., Dorfman I. (1980). Hamiltonian operators and algebraic structures related to them. Funct. Anal. Appl. 13:248–262MATHCrossRefGoogle Scholar
  8. 8.
    Magnin, L.: Adjoint and Trivial Cohomology Tables for Indecomposable Nilpotent Lie Algebras of Dimension ≤ 7 over C. e-book, 1–906 (1995).Google Scholar
  9. 9.
    Scheuneman, J.: Affine structures on three-step nilpotent Lie algebras. Proc. Amer. Math. Soc. 46, (1974).Google Scholar
  10. 10.
    Segal D. (1992). The structure of complete left–symmetric algebras. Math. Ann. 293:569–578MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Semenov- Tian-Shansky M.A. (1983). What is a classical r-matrix. Funct. Anal. Appl. 17:259–272CrossRefGoogle Scholar
  12. 12.
    Zelmanov E. (1987). On a class of local translation invariant Lie algebras. Soviet Math. Dokl. 35:216–218Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität WienViennaAustria

Personalised recommendations