Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 048, 10 pages      math.RT/0605091      http://dx.doi.org/10.3842/SIGMA.2006.048

On Deformations and Contractions of Lie Algebras

Alice Fialowski a and Marc de Montigny b
a) Institute of Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary
b) Campus Saint-Jean and Theoretical Physics Institute, University of Alberta, 8406 - 91 Street, Edmonton, Alberta, T6C 4G9, Canada

Received February 24, 2006, in final form April 25, 2006; Published online May 03, 2006

Abstract
In this contributed presentation, we discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras. We suggest that with appropriate combinations of both procedures one may construct new Lie algebras. We first discuss low-dimensional Lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation, the converse is not true in general. Also we note that some Lie algebras belonging to parameterized families are singled out by the irreversibility of deformations and contractions. After reminding that global deformations of the Witt, Virasoro, and affine Kac-Moody algebras allow one to retrieve Lie algebras of Krichever-Novikov type, we contract the latter to find new infinite dimensional Lie algebras.

Key words: Lie algebras; deformations; contractions; Kac-Moody algebras.

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References

  1. Fialowski A., de Montigny M., Contractions and deformations of Lie algebras, J. Phys. A: Math. Gen., 2005, V.38, 6335-6349.
  2. Gerstenhaber M., On the deformation of rings and algebras, Ann. Math., 1964, V.79, 59-103;
    Gerstenhaber M., On the deformation of rings and algebras II, Ann. Math., 1966, V.84, 1-19;
    Gerstenhaber M., On the deformation of rings and algebras III, Ann. Math., 1968, V.88, 1-34;
    Gerstenhaber M., On the deformation of rings and algebras IV, 1974, V.99, 257-276;
    Nijenhuis A., Richardson R.W., Deformations of Lie algebra structures, J. Math. Mech., 1967, V.17, 89-105;
    Fialowski A., Deformations of Lie algebras, Math. USSR Sbornik, 1986, V.55, 467-472;
    Fialowski A., An example of formal deformations of Lie algebras, in Proceedings of NATO Conference on Deformation Theory of Algebras and Applications, Editors M. Hazawinkel and M. Gerstenhaber, Dordrecht, Kluwer, 1988, 375-401;
    Fialowski A., Fuchs D., Construction of miniversal deformations of Lie algebras, J. Funct. Anal., 1999, V.161, 76-110, math.RT/0006117.
  3. Fialowski A., Schlichenmaier M., Global deformations of the Witt algebra of Krichever-Novikov type, Commun. Contemp. Math., 2003, V.5, 921-945, math.QA/0206114;
    Fialowski A., Schlichenmaier M., Global geometric deformations of current algebras as Krichever-Novikov type algebras, Comm. Math. Phys., 2005, V.260, 579-612, math.QA/0412113.
  4. Inönü E., Wigner E.P., On the contraction of groups and their representations, Proc. Nat. Acad. Sci. U.S.A., 1953, V.39, 510-524;
    Saletan E., Contraction of Lie groups, J. Math. Phys., 1961, V.2, 1-21;
    Gilmore R., Lie groups, Lie algebras, and some of their applications, New York, Wiley, 1974, Chapter 10;
    Talman J.D., Special functions: a group theoretic approach, New York, Benjamin, 1968.
  5. Onishchik A.L., Vinberg E.B., Lie groups and Lie algebras, Enclycopaedia of Mathematical Sciences, Vol. 41, Berlin, Springer, 1991, Chapter 7.
  6. Weimar-Woods E., Contractions, generalized Inönü-Wigner contractions and deformations of finite-dimensional Lie algebras, Rev. Math. Phys., 2000, V.12, 1505-1529;
    Lévy-Nahas M., Deformation and contraction of Lie algebras, J. Math. Phys., 1967, V.8, 1211-1222;
    Lõhmus J., Tammelo R., Contractions and deformations of space-time algebras I. General theory and kinematical algebras, Hadronic J., 1997, V.20, 361-416;
    Fialowski A., O'Halloran J., A comparison of deformations and orbit closure, Comm. Algebra, 1990, V.18, 4121-4140.
  7. Conatser C.W., Contractions of the low-dimensional real Lie algebras, J. Math. Phys., 1972, V.13, 196-203.
  8. Burde D., Steinhoff C., Classification of orbit closures of 4-dimensional complex Lie algebras, J. Algebra, 1999, V.214, 729-739.
  9. Fialowski A., Deformations of some infinite-dimensional Lie algebras, J. Math. Phys., 1990, V.31, 1340-1343.
  10. Lecomte P.B.A., Roger C., Rigidity of current Lie algebras of complex simple type, J. London Math. Soc. (2), 1988, V.37, 232-240.
  11. Fialowski A., Penkava M., Versal deformations of three-dimensional Lie algebras as L algebras, Commun. Contemp. Math., 2005, V.7, 145-165, math.RT/0303346.
  12. Goddard P., Olive D., Kac-Moody and Virasoro algebras, New York, World Scientific, 1988;
    Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory, New York, Springer, 1997;
    Tsvelik A.M., Quantum field theory in condensed matter physics, Cambridge Univ. Press, 2003.
  13. Krichever I.M., Novikov S.P., Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons, Funct. Anal. Appl., 1987, V.21, 126-142;
    Krichever I.M., Novikov S.P., Virasoro-type algebras, Riemann surfaces and strings in Minkowski space, Funct. Anal. Appl., 1987, V.21, 294-307;
    Krichever I.M., Novikov S.P., Algebras of Virasoro type, energy-momentum tensor and decomposition operators on Riemann surfaces, Funct. Anal. Appl., 1989, V.23, 19-33.
  14. Majumdar P., Inönü-Wigner contraction of Kac-Moody algebras, J. Math. Phys., 1993, V.34, 2059-2065, hep-th/9207057;
    Olive D.I., Rabinovici E., Schwimmer A., A class of string backgrounds as a semiclassical limit of WZW models, Phys. Lett. B, 1994, V.321, 361-364, hep-th/9311081.

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