Yurii A. Neretin
We construct some canonically defined central extensions of groups of symplectomorphisms. We show that this central extension is nontrivial in the case of a torus of dimension $\ge 6$ and in the case of a two-dimensional surface of genus $\ge 3$.
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Centralizers of Hamiltonian circle actions on rational ruled surfaces
本文研究两类有理直纹面上 Hamilton $S^1$ 作用的等变辛微分同胚群，通过近复结构分层及 Delzant、Karshon 分类确定其同伦型为环面或两环面的同伦推出。
By a theorem of Banyaga the group of diffeomorphisms of a manifold $P$ preserving a regular contact form $α$ is a central $S^1$ extension of the commutator of the group of symplectomorphisms of the base $B = P/S^1$. We show that if $T$ is a Hamiltonian maximal torus in the group of symplectomorphism of $B$, then its preimage under the extension map is a maximal torus not only in the group $\Diff(P, α)$ of diffeomorphisms of $P$ preserving $α$ but also in the much bigger group of contactomorphisms $\Diff (P, ξ)$, the group of diffeomorphism of $P$ preserving the contact distribution $ξ= \ker α$…
This paper concerns the action of linear symplectomorphisms on linear symplectic forms by conjugation in even dimensions. We prove that pfaffian and $-\frac{1}{2}\operatorname{tr}(JA)$ (sum function) of $A$ are invariants on the action. We use these invariants to provide a complete description of the orbit space in dimension four. In addition, we investigate the geometric shapes of the individual orbits in dimension four. In symplectic geometry, our classification result in dimension four provides a necessary condition for two symplectic forms on $\mathbb{R}^{4}$ to be intertwined by symplecto…
Norm-squared of the momentum map in infinite dimensions with applications to Kähler geometry and symplectic connections
本研究率先在无穷维中以动量映射范数平方为工具，推导其临界点Hessian性质与稳定子分解，应用于辛微分同胚群作用的两类空间，推广Calabi结果并发现非等变带来的新现象，还通过预量子化得到辛微分同胚群的新中心扩张。
We determine the universal central extension of the Lie algebra of hamiltonian vector fields, thereby classifying its central extensions. Furthermore, we classify the central extensions of the Lie algebra of symplectic vector fields, of the Poisson Lie algebra, and of its compactly supported version.
We present a geometric construction of central extensions of covering groups of the group of volume preserving diffeomorphisms, integrating central extensions of the Lie algebra of divergence free vector fields defined by Lichnerowicz cocycles. Certain covering spaces of non-linear Grassmannians can be realized as prequantizable coadjoint orbits in these Lie group extensions.
In this review paper, we present several results on central extensions of the Lie algebra of symplectic (Hamiltonian) vector fields, and compare them to similar results for the Lie algebra of (exact) divergence free vector fields. In particular, we comment on universal central extensions and integrability to the group level.
We present a geometric construction of central S^1-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie algebra. We use this to find nontrivial central S^1-extensions of the universal cover of the group of Hamiltonian diffeomorphisms. In the process, we obtain central S^1-extensions of Lie groups that act by exact strict contact transformations.
In this paper, we introduce a notion of a central $U(1)$-extension of a double Lie groupoid and show that it defines a cocycle in the certain triple complex.
Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds
本文研究李代数中心扩张与动力系统的联系，分析幂零李代数无限中心扩张序列的余伴随表示轨道，结合紧幂零流形证明辛幂零流形的覆盖李群作为可解李群可具有任意秩。
We give a characterisation of central extensions of a Lie group G by the non-zero complex numbers in terms of a differential two-form on G and a differential one-form on GxG. This is applied to the case of the central extension of the loop group.
Universal central extension of the Lie algebra of exact divergence-free vector fields
本文通过分析精确无散度向量场背后的Leibniz代数，构造了其李代数的万有中心扩张，证明了Roger 1995年的猜想，并应用于构造紧致三维流形上相应无限维李群的万有中心扩张。
The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π_2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integra…
Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a new connective structure on the lifting gerbe associated to this problem. Our main result classifies all connections on the central extension of a given principal bundle. In particular, we find that admissible connections are in one-to-one correspondence with parallel trivializations of the lifting gerbe. Moreover, we prove a vanishing result for Neeb's obstruction classes for finite-dimensional Lie groups.
Motivated by positive energy representations, we classify those continuous central extensions of the compactly supported gauge Lie algebra that are covariant under a 1-parameter group of transformations of the base manifold.
Smooth contact lifts to central extensions of Carnot groups
研究Carnot群间光滑切触映射向其中心扩张间映射的提升存在性问题，给出基于de Rham势拉回Rumin微分的充要条件及Pansu拉回的必要条件，并证明后者在特定条件下充分。
We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed in [NW09], is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting.
Contact lifts and Holder lifts to central extension of Carnot groups
本文研究了Carnot群中映射提升的存在性与刚性，证明了Lipschitz和Sobolev映射的接触提升存在，以及准共形映射接触提升的双Lipschitz性质。
We briefly indicate some implications of [1] for the second Lie algebra cohomology of equivariant map algebras and (twisted multi) loop algebras.
Central extensions for loop groups of area-preserving diffeomorphisms and their fuzzy sphere limits
研究面积保持微分同胚环的中心扩张，并揭示其在大k极限下的模糊球极限性质。
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