Skip to main content
SpringerLink
Log in

Classical r-Matrices and Novikov Algebras

  • Original Article
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Balinskii A.A., Novikov S.P. (1985). Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras. Soviet Math. Dokl. 32:228–231

    Google Scholar 

  2. Benoist Y. (1995). Une nilvariété non affine. J. Differential Geom. 41:21–52

    MATH  MathSciNet  Google Scholar 

  3. Burde D. (1996). Affine structures on nilmanifolds. Int. J. Math. 7:599–616

    Article  MATH  MathSciNet  Google Scholar 

  4. Burde, D., Dekimpe, K.: Novikov structures on solvable Lie algebras. J. Geom. Phys. (2006) (to appear)

  5. Dorfman I., Gel’fand I.M. (1979). Hamiltonian operators and algebraic structures associated with them. Funktsional. Anal. i Prilozhen. 13:13–30

    MATH  MathSciNet  Google Scholar 

  6. Frenkel I., Huang Y.Z., Lepowsky L. (1993). On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 494:1–64

    MathSciNet  Google Scholar 

  7. Gel’fand I.M., Dorfman I. (1980). Hamiltonian operators and algebraic structures related to them. Funct. Anal. Appl. 13:248–262

    Article  MATH  Google Scholar 

  8. Magnin, L.: Adjoint and Trivial Cohomology Tables for Indecomposable Nilpotent Lie Algebras of Dimension ≤ 7 over C. e-book, 1–906 (1995).

  9. Scheuneman, J.: Affine structures on three-step nilpotent Lie algebras. Proc. Amer. Math. Soc. 46, (1974).

  10. Segal D. (1992). The structure of complete left–symmetric algebras. Math. Ann. 293:569–578

    Article  MATH  MathSciNet  Google Scholar 

  11. Semenov- Tian-Shansky M.A. (1983). What is a classical r-matrix. Funct. Anal. Appl. 17:259–272

    Article  Google Scholar 

  12. Zelmanov E. (1987). On a class of local translation invariant Lie algebras. Soviet Math. Dokl. 35:216–218

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Nordbergstr. 15, 1090, Vienna, Austria

    Dietrich Burde

Authors
  1. Dietrich Burde
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dietrich Burde.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burde, D. Classical r-Matrices and Novikov Algebras. Geom Dedicata 122, 145–157 (2006). https://doi.org/10.1007/s10711-006-9059-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-006-9059-y

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation