Geometriae Dedicata

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

Classical r-Matrices and Novikov Algebras

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 


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  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

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