65 research outputs found
Formal deformations, contractions and moduli spaces of Lie algebras
Jump deformations and contractions of Lie algebras are inverse concepts, but
the approaches to their computations are quite different. In this paper, we
contrast the two approaches, showing how to compute jump deformations from the
miniversal deformation of a Lie algebra, and thus arrive at the contractions.
We also compute contractions directly. We use the moduli spaces of real
3-dimensional and complex 3 and 4-dimensional Lie algebras as models for
explaining a deformation theory approach to computation of contractions.Comment: 27 page
Leibniz algebra deformations of a Lie algebra
In this note we compute Leibniz algebra deformations of the 3-dimensional
nilpotent Lie algebra and compare it with its Lie
deformations. It turns out that there are 3 extra Leibniz deformations. We also
describe the versal Leibniz deformation of with the versal
base.Comment: 15 page
Stratification of moduli spaces of Lie algebras, similar matrices and bilinear forms
In this paper, the authors apply a stratification of moduli spaces of complex Lie algebras to analyzing the moduli spaces of nxn matrices under scalar similarity and bilinear forms under the cogredient action. For similar matrices, we give a complete description of a stratification of the space by some very simple projective orbifolds of the form P^n/G, where G is a subgroup of the symmetric group sigma_{n+1} acting on P^n by permuting the projective coordinates. For bilinear forms, we give a similar stratification up to dimension 4
Versal deformation of the Lie algebra
We investigate deformations of the infinite dimensional vector field Lie algebra spanned by the fields , where . The goal is to describe the base of a ``versal'' deformation; such a versal deformation induces all the other nonequivalent deformations and solves the deformation problem completely. \u
Global Geometric Deformations of the Virasoro algebra, current and affine algebras by Krichever-Novikov type algebra
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.Comment: 17 page
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