155 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
Global geometric deformations of current algebras as Krichever-Novikov type algebras
We construct algebraic-geometric families of genus one (i.e. elliptic)
current and affine Lie algebras of Krichever-Novikov type. These families
deform the classical current, respectively affine Kac-Moody Lie algebras. The
construction is induced by the geometric process of degenerating the elliptic
curve to singular cubics. If the finite-dimensional Lie algebra defining the
infinite dimensional current algebra is simple then, even if restricted to
local families, the constructed families are non-equivalent to the trivial
family. In particular, we show that the current algebra is geometrically not
rigid, despite its formal rigidity. This shows that in the infinite-dimensional
Lie algebra case the relations between geometric deformations, formal
deformations and Lie algebra two-cohomology are not that close as in the
finite-dimensional case. The constructed families are e.g. of relevance in the
global operator approach to the Wess-Zumino-Witten-Novikov models appearing in
the quantization of Conformal Field Theory.Comment: 35 pages, AMS-Late
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
Cohomology and deformations of the infinite dimensional filiform Lie algebra m_0
Denote m_0 the infinite dimensional N-graded Lie algebra defined by basis
e_i, i>= 1 and relations [e_1,e_i] = e_(i+1) for all i>=2. We compute in this
article the bracket structure on H1(m_0,m_0), H2(m_0,m_0) and in relation to
this, we establish that there are only finitely many true deformations of m_0
in each nonpositive weight, by constructing them explicitely. It turns out that
in weight 0 one gets exactly the other two filiform Lie algebras.Comment: 25 page
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