Deformation theory via differential graded Lie algebras
英文摘要:
It is a basic introduction to differential graded Lie algebras, Maurer-Cartan equation and associated deformation functors.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
Marco Manetti
It is a basic introduction to differential graded Lie algebras, Maurer-Cartan equation and associated deformation functors.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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We construct the deformation functor associated to a couple of morphisms of differential graded Lie algebras, and use it to study the infinitesimal deformations of a holomorphic map of compact complex manifolds. In particular, in the case of Kähler manifolds we describe a generalizzation of the semiregularity map.
We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf F are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf End(E), where E is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to deformations of F is Ext^1(F,F), and obstructions are contained in Ext^2(F,F). The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA whose cohomology is concentrated in nonnegative degrees …
For a smooth Deligne-Mumford stack over $\CC$, we define its associated Kodaira-Spencer differential graded Lie algebra and show that the deformation functor of the stack is isomorphic to the deformation functor of the Kodaira-Spencer algebra if the stack is proper over $\CC$.
We construct a differential graded Lie algebra $\fg$ controlling the Poisson deformations of an affine Poisson variety. We analyse $\fg$ in the case of affine Gorenstein toric Poisson varieties. Moreover, explicit description of the second and third Hochschild cohomology groups is given for three-dimensional affine Gorenstein toric varieties.
In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and present the construction for (non-graded) differential Lie algebras. Then, we define the category of differential graded Lie groups and study its properties. We show how to associate a differential graded Lie algebra to every differential graded Lie group and vice-versa. For the DGLA $\to$ DGLG direction, the main ``tools'' are graded Hopf algebras and Harish-Chandra pairs (HCP) -- we defi…
We describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any $n$-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra $F_n^3(0)$. We establish that in the same way any $n$-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras $F_{n}^1,$ $F_{n}^2$ and $F_{n}^3(α)$. Moreover, we describe the linear integrable deformations of above-mentioned algebras with a fixed basis of $HL^2$ in the set of all $n$-dimensional Leibni…
Character sheaves for classical graded Lie algebras
本文针对源自特殊线性群内自同构的分次李代数与Vinberg II型典型分次李代数,开展其特征层的研究工作。
This paper is devoted to deformation theory of graded Lie algebras over $\Z$ or $\Z_l$ with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artin local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic deformations usi…
Cuspidal character sheaves on graded Lie algebras II
本文研究了有级数李代数上尖点特征sheaves的分类及其支持,为明确分类这些sheaves提供了重要进展。
Relating depth graded and block graded motivic Lie algebras
本文以块滤过实现余根滤过,建立块分次与全奇多重 zeta 值之间的字典,刻画深度分次动机李代数关系,并探讨非均匀 Broadhurst–Kreimer 猜想。
Consider the neutral Tannakian category mixed Tate motives over Z, in this paper we suggest a way to understand the structure of depth-graded motivic Lie subalgebra generated by the depth one part. We will show that from an isomorphism conjecture proposed by K. Tasaka we can deduce the F. Brown matrix conjecture and the non-degenerated conjecture about depth-graded motivic Lie subalgebra generated by the depth one part.
Invariant systems and character sheaves for graded Lie algebras
该研究通过引入微分方程不变系统构造分次李代数上的特征层,并与既有结果共同给出Vinberg I型经典情形的全部尖点特征层。
To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman-Millson, Green-Lazarsfeld, Nadel, Simpson, Dimca-Papadima, and of the second author.
Cuspidal character sheaves on graded Lie algebras
本文通过邻近循环构造和傅里叶-萨托变换,完成了文金类型I分层经典李代数极小特征特征sheaves的分类。
This text gives a construction of a differential graded Lie algebra in Nori's category of effective homological motives. In fact the construction works in more a general setting than that of an Abelian category. This allows us to give the rational homotopy Lie algebra of a 1-connected space a motivic structure. As a consequence the rational homotopy Lie algebra inherits a mixed Hodge structure and Galois module structure.
Character Sheaves for Graded Lie Algebras: Stable Gradings
本研究针对稳定分次李代数,通过分析[GVX2]中邻近周期层的傅里叶变换,构造出推测为尖点特征层的满支撑特征层,涉及单位根处复反射群相关Hecke代数的不可约表示。
Integration à la Harish-Chandra for Bi-Graded Lie Algebras
本文研究了$\mathbb Z_2 \times \mathbb Z_2$双分级李代数的性质,并探讨了基于哈里-恰特拉奇对的李群-代数对应关系及其应用。
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