Deformations of filiform Lie algebras and superalgebras (original) (raw)
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Infinitesimal deformations of the Lie superalgebra Ln,m
Journal of Geometry and Physics, 2008
In this paper, we continue the study of the infinitesimal deformations of the Lie superalgebra L n,m that we have started in [M. Bordemann, J.R. Gómez, Yu. Khakimdjanov, R.M. Navarro, Some deformations of nilpotent Lie superalgebras, J. Geom. Phys. 57 (2007) 1391-1403]. These deformations allow us to obtain all filiform Lie superalgebras. In [M. Bordemann, J.R. Gómez, Yu. Khakimdjanov, R.M. Navarro, Some deformations of nilpotent Lie superalgebras, J. Geom. Phys. 57 -1403], we gave a method that allows us to determine the dimension of the space of deformations of type Hom(S 2 (L n,m 1 ), L n,m 0 ) and we calculated a basis of the aforementioned space of deformations for n ≥ 2m − 1. In this paper, we conclude the study by developing a method to calculate a basis of the space of deformations Hom(S 2 (L n,m 1 ), L n,m 0 ) for the rest of possibilities n < 2m − 1. We particularize for even n and also give an algorithm for computing a cocycle basis for the given concrete dimensions n and m.
Some deformations of nilpotent Lie superalgebras
Journal of Geometry and Physics, 2007
In this paper we study the infinitesimal deformations of the Lie superalgebra L n,m . By means of these deformations all filiform Lie superalgebras can be obtained. In particular, we give a method that will allow us to determine the dimension of the space of deformations of type Hom(S 2 (L n,m 1 ), L n,m 0 ). Note that this type of deformation is the only one that occurs for Lie superalgebras which are not Lie algebras. Furthermore we develop a method for calculating a basis of the aforementioned space of deformations Hom(S 2 (L n,m 1 ), L n,m 0 ), giving it explicitly for n ≥ 2m − 1.
A complete description of all the infinitesimal deformations of the Lie superalgebra
2010
In this paper, we find the dimension and a method to obtain a basis of some infinitesimal deformations of the model Lie superalgebra L n,m . These deformations lie in Hom(L n ∧V 1 , V 1 ) being L n and V 1 the even and odd parts of the Lie superalgebra L n,m respectively. As L n corresponds to the model filiform Lie algebra, then these deformations can also be identified with the space of the infinitesimal deformations of the filiform L n -module V 1 . Combining with Bordemann et al. (2007) [2], Gómez et al. (2008) [3] and Khakimdjanov and Navarro [4] we therefore obtain a complete classification of all the infinitesimal deformations of the model Lie superalgebra L n,m .
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.
A class of quadratic deformations of Lie superalgebras
2010
We study certain Z_2-graded, finite-dimensional polynomial algebras of degree 2 which are a special class of deformations of Lie superalgebras, which we call quadratic Lie superalgebras. Starting from the formal definition, we discuss the generalised Jacobi relations in the context of the Koszul property, and give a proof of the PBW basis theorem. We give several concrete examples of quadratic Lie superalgebras for low dimensional cases, and discuss aspects of their structure constants for the `type I' class. We derive the equivalent of the Kac module construction for typical and atypical modules, and a related direct construction of irreducible modules due to Gould. We investigate in detail one specific case, the quadratic generalisation gl_2(n/1) of the Lie superalgebra sl(n/1). We formulate the general atypicality conditions at level 1, and present an analysis of zero-and one-step atypical modules for a certain family of Kac modules.
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).
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.
Cohomology of Lie superalgebras and their generalizations
Journal of Mathematical Physics, 1998
The cohomology groups of Lie superalgebras and, more generally, of ε Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is nontrivial. Two general propositions are proved, which help to calculate the cohomology groups. Several examples are included to show the peculiarities of the super case. For L=sl(1|2), the cohomology groups H1(L,V) and H2(L,V), with V a finite-dimensional simple graded L-module, are determined, and the result is used to show that H2(L,U(L)) [with U(L) the enveloping algebra of L] is trivial. This implies that the superalgebra U(L) does not admit any nontrivial formal deformations (in the sense of Gerstenhaber). Garland’s theory of universal central extensions of Lie algebras is generalized to the case of ε Lie algebras.
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.