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Versal Deformations of Leibniz Algebras

Published online by Cambridge University Press:  03 June 2008

Alice Fialowski ,
Ashis Mandal  and
Goutam Mukherjee
Alice Fialowski
Affiliation:
Eötvös Loránd University, Budapest, Hungary, fialowsk@cs.elte.hu.
Ashis Mandal
Affiliation:
Indian Statistical Institute, Kolkata, India, ashis_r@isical.ac.in.
Goutam Mukherjee
Affiliation:
Indian Statistical Institute, Kolkata, India, goutam@isical.ac.in
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Abstract

In this work we consider deformations of Leibniz algebras over a field of characteristic zero. The main problem in deformation theory is to describe all non-equivalent deformations of a given object. We give a method to solve this problem completely, namely work out a construction of a versal deformation for a given Leibniz algebra, which induces all non-equivalent deformations and is unique on the infinitesimal level.

Keywords

Leibniz algebraLeibniz cohomologyinfinitesimal deformationversal deformationobstruction
Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 2 , April 2009 , pp. 327 - 358
Copyright
Copyright © ISOPP 2009

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References

1.Albeverio, S., Ayupov, Sh. A. and Omirov, B.A.: On nilpotent and simple Leibniz algebras, Comm. Alg. 33 (2005), 159–172CrossRefGoogle Scholar
2.Albeverio, S., Omirov, B.A. and Rakhimov, I.S.: Varieties of nilpotent complex Leibniz algebras of dimension less than five, Comm. Alg. 33 (2005), 1575–1585CrossRefGoogle Scholar
3.Ayupov, Sh. A. and Omirov, B.A.: On some classes of nilpotent Leibniz algebras, Siberian Math. Journal (1) 42 (2001), 18–29CrossRefGoogle Scholar
4.Balavoine, D.: Deformations of algebras over a quadratic operad, Contemporary Maths. AMS 202 (1997), 207–234CrossRefGoogle Scholar
5.Fialowski, A.: An example of formal deformations of Lie algebras, “NATO Conference on deformation theory of algebras and applications, Proceedings”,Kluwer,Dordrecht(1988)375–401Google Scholar
6.Fialowski, A. and Fuchs, D.: Construction of miniversal Deformations of Lie Algebras, Journal of Functional Analysis 161 (1999), 76–110CrossRefGoogle Scholar
7.Fialowski, A. and Penkava, M.: Versal Deformations of Three-dimensional Lie Algebras as L ∞ algebras, Comm. Cont. Math. 7 (2005), 145–165CrossRefGoogle Scholar
8.Gerstenhaber, M.: On the Deformations of Rings and Algebras, Ann. of Math. 79 (1964), 59–103CrossRefGoogle Scholar
9.Gerstenhaber, M.: On the Deformations of Rings and Algebras, Ann. of Math. 84 (1966), 1–19CrossRefGoogle Scholar
10.Gerstenhaber, M.: On the Deformations of Rings and Algebras, Ann. of Math. 88 (1968), 1–34CrossRefGoogle Scholar
11.Gerstenhaber, M.: On the Deformations of Rings and Algebras, Ann. of Math. 99 (1974), 257–276CrossRefGoogle Scholar
12.Gerstenhaber, M.: The Cohomology Structure of an Associative Ring, Ann. of Math. 78 (1963), 267–288CrossRefGoogle Scholar
13.Goldman, W.M. and Millson, J.J.: The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Etudes Sci. Pub. Math. 67 (1988), 43–96CrossRefGoogle Scholar
14.Harrison, D. K.: Commutative algebras and cohomology, Trans. Amer. Math. Soc. 104 (1962), 191–204CrossRefGoogle Scholar
15.Hartshorne, R.: “Algebraic Geometry”, Springer-Verlag, Berlin/New York, 1977CrossRefGoogle Scholar
16.Illusie, L.: “Complexe cotangent et deformations I”, Lecture Notes in Math. 239, Springer- Verlag, Berlin/New York, 1971Google Scholar
17.Kontsevich, M.: “Topics in Algebra: Deformation Theory”, Lecture Notes, Univ. of California Press, Berkeley, CA, 1994Google Scholar
18.Laudal, O.A.: “Formal moduli of algebraic structures”, Lecture Notes in Math. 754, Springer-Verlag, Berlin/New York, 1979Google Scholar
19.Loday, J.-L.: Une version non commutative des algèbres de Lie: les algebrès de Leibniz, Enseign. Math. 39 (2), No.3–4 (1993), 269–293Google Scholar
20.Loday, J.-L.: Overview on Leibniz algebras, dialgebras and their homology, Fields Institute Communications 17 (1997), 91–102Google Scholar
21.Loday, J.-L.: Dialgebras and related operads, Lecture Notes in Math. 1763 (2001), 7–66Google Scholar
22.Loday, J.-L. and Pirashvili, T.: Universal enveloping algebras of Leibniz algebras and (co)homology, Math.Ann. 296 (1993), 139–158CrossRefGoogle Scholar
23.Mandal, A.: An Example of Constructing Versal Deformation for Leibniz Algebras, To appear in Comm. Alg., arXiv : math.QA/071212.2096v1 13 Dec 2007Google Scholar
24.Palamodov, V.P.: Deformations of complex spaces, Russian Math. Surveys 31 (1976)CrossRefGoogle Scholar
25.Schlessinger, M.: Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208–222CrossRefGoogle Scholar