On deformations and contractions of Lie algebras (original) (raw)
Related papers
Deformations and contractions of Lie algebras
Journal of Physics A: Mathematical and General, 2005
We discuss the mutually opposite procedures of deformations and contractions of Lie algebras. Our main purpose is to illustrate the fact that, with appropriate combinations of both procedures, we obtain new Lie algebras. Firstly, we discuss low-dimensional Lie algebras, and these simple examples illustrate that, whereas for every contraction there exists a reverse deformation, the converse is not true in general. We point out that otherwise ordinary members of parameterized families of Lie algebras are singled out by this irreversibility of deformations and contractions. Then, we remind that global deformations of the Witt, Virasoro, and affine Kac-Moody algebras allow one to retrieve Lie algebras of Krichever-Novikov type and show that, in turn, contractions of the latter lead to new infinite dimensional Lie algebras.
Deformations of some infinite-dimensional Lie algebras
Journal of Mathematical Physics, 1990
The concept of a versal deformation of a Lie algebra is investigated and obstructions to extending an infinitesimal deformation to a higher-order one are described. The rigidity of the Witt algebra and the Virasoro algebra is deduced from cohomology computations for certain Lie algebras of vector fields on the real line. The Lie algebra of vector fields on the line that vanish at the origin also turns out to be rigid. All the affine Lie algebras are rigid; this is derived from the cohomology of their maximal nilpotent subalgebra. On the other hand, the maximal nilpotent subalgebras in both the Virasoro and affine cases are not rigid and have interesting nontrivial deformations (in fact, most vector field Lie algebras are not rigid).
Global Geometric Deformations of Current Algebras as Krichever-Novikov Type Algebras
Communications in Mathematical Physics, 2005
In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever-Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac-Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. 1 2 A. FIALOWSKI AND M. SCHLICHENMAIER, FEBRUARY 2, 2008
International Journal of Theoretical Physics, 2007
In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever-Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac-Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. 1 2 A. FIALOWSKI AND M. SCHLICHENMAIER, FEBRUARY 2, 2008
1985
The author considers general questions of deformations of Lie algebras over a field of characteristic zero, and the related problems of computing cohomology with coefficients in adjoint representations. The construction of a versal family, and the construction of obstructions to the extension of deformations, are also considered. Bibliography: 13 titles.
Deformation of the Lie Algebra L
2001
In the last decade the interest in deformation theory has grown in many areas of mathematics and physics. The deformation question is completely solved by describing a ‘‘versal’’ deformation of the given object; such a deformation induces all the other deformations. This problem turns out to be hard and a general procedure for solving this was given only recently in FF2 for Lie algebras. It is still not trivial to apply this construction to specific examples.
Deformations of Lie algebras. Math.USSR-Sb. 55 (1986), 467-473
The author considers general questions of deformations of Lie algebras over a field of characteristic zero, and the related problems of computing cohomology with coefficients in adjoint representations. The construction of a versal family, and the construction of obstructions to the extension of deformations, are also considered. Bibliography: 13 titles.
Deformations of Lie algebras using σ-derivations
Journal of Algebra, 2006
In this article we develop an approach to deformations of the Witt and Virasoro algebras based on σ-derivations. We show that σ-twisted Jacobi type identity holds for generators of such deformations. For the σ-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the q-deformations of the Witt and Virasoro algebras associated to qdifference operators, providing also corresponding q-deformed Jacobi identities. 1