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1
Content available remote

An extention of Nomizu’s Theorem –A user’s guide–

100%
Kasuya H.
Complex Manifolds
|
2016
|
tom 3
|
nr 1
EN
For a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ\G, C) of the solvmanifold Γ\G. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ\G, C).
2
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Structures ofW(2.2) Lie conformal algebra

80%
Yuan L., Wu H.
Open Mathematics
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2016
|
tom 14
|
nr 1
629-640
EN
The purpose of this paper is to study W(2, 2) Lie conformal algebra, which has a free ℂ[∂]-basis {L, M} such that [...] [LλL]=(∂+2λ)L,[LλM]=(∂+2λ)M,[MλM]=0 $\begin{equation}[{L_\lambda }L] = (\partial + 2\lambda )L,[{L_\lambda }M] = (\partial + 2\lambda )M,[{M_\lambda }M] = 0]\end{equation}$ . In this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.
3
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A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

80%
Poon Y. S., Simanyi J.
Complex Manifolds
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2017
|
tom 4
|
nr 1
137-154
EN
A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical მ̄-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex manifolds are nilmanifolds with an abelian complex structure. For a particular class of holomorphic Poisson structures, this result leads to a Hodge-type decomposition of the holomorphic Poisson cohomology. We provide examples when the nilmanifolds are 2-step.
4
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Jacobi-Bernoulli cohomology and deformations of schemes and maps

70%
Ran Z.
Open Mathematics
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2012
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tom 10
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nr 4
1541-1591
EN
We introduce a notion of Jacobi-Bernoulli cohomology associated to a semi-simplicial Lie algebra (SELA). For an algebraic scheme X over ℂ, we construct a tangent SELA J X and show that the Jacobi-Bernoulli cohomology of J X is related to infinitesimal deformations of X.
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