arXiv research

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 …

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.

Criterion for lifting smooth contact maps between Carnot groups to central extensions.

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 $\ge 3$.

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 preq…

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 o…

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.

The central extension of the Thompson group $T$ that arises in the quantized Teichmüller theory is 12 times the Euler class. This extension is obtained by taking a (partial) abelianization of the so-called braided Ptolemy-Thompson group introduced and studied in \cite{FK2}. We describe then the cyclic central extension…

The paper derives the QGS equations using stochastic central extensions.

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.

Simple construction of Lie 2-groups from loop 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 …

Researchers solve a 25-year-old conjecture about vector fields.

Compute central extension of mapping class group from stated skein algebra

Classifies central extensions for area-preserving diffeomorphisms and shows they are fuzzy sphere limits.

Quantization of universal Teichmüller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group $T$. This yields certain central extensions of $T$ by $\mathbb{Z}$, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central ext…

New topological Riemann-Roch theorem for circle fibrations.

New Virasoro-like structures for circle diffeomorphisms with breaks.

Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.

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.

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.

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 …

A central extension of the loop group of a Lie group is called transgressive, if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the Lie group. Transgressive loop group extensions are those that can be explored by finite-dimensional, higher-categorical geometry ov…

The central extension of mapping class groups of punctured surfaces of finite type that arises in Chekhov-Fock quantization is 12 times of the Meyer class plus the Euler classes of the punctures, which agree with the one arising in the Kashaev quantization.

When a Lie group $G$ has a central $U(1)$-extension, there is a cocycle in the simplicial de Rham complex $Ω^3(NG)$ which represents the Dixmier-Douady class. Mickelsson and Brylinski, McLaughlin constructed a central $U(1)$-extension $\widehat{LSU(2)} \rightarrow LSU(2)$ whose Dixmier-Douady class in $Ω^3(NLSU(2))$ is…

Let $D$ be a 2-dimensional closed unit disk and $\rm{Symp}(D,0)_{\rm{rel}}$ the group of symplectomorphisms preserving the origin and the boundary $\partial D$ pointwise. We consider the $\mathbb{R}$-valued flux homomorphism on $\rm{Symp}(D,0)_{\rm{rel}}$ and define the central $\mathbb{R}$-extension called the $\mathb…

In "The Yang-Mills equations over Riemann surfaces", Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory. We generalize their study to all closed, compact, connected, possibly nonorientable surfaces. We introduce the notion of "super central extension" of the fund…

New group found not satisfying quasi-isometric triviality property.

Computes quandle associated groups using group homology.

Let $M$ be a manifold with a closed, integral $(k+1)$-form $ω$, and let $G$ be a Fréchet-Lie group acting on $(M,ω)$. As a generalization of the Kostant-Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of $\mathfrak{g}$ by $\mathbb{R}$, indexed by $H^{k-1}(M,\mathbb{R})^*$…

New knot quandle structure for twist-spun trefoils discovered.

The paper studies twisted Alexander polynomials for knot groups in various extensions.

This paper is a rigorous study of the dual pair structure of the ideal fluid and the dual pair structure for the $n$-dimensional Camassa-Holm (EPDiff) equation, including the proofs of the necessary transitivity results. In the case of the ideal fluid, we show that a careful definition of the momentum maps leads natura…

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the relationship between $S^1$-gerbes and groupoid $S^1$-central extensions. We define connections and curvings for groupoid $S^1$-cent…

This paper develops an approach for describing centrally extended groups, as determining the adjoint groups associated with quandles. Furthermore, we explicitly describe such groups of some quandles. As a corollary, we determine some second quandle homologies.

We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid $G$-extensions, which we call "connections on gerbes", and study the induced connections on various associated bundles. We also prove analogues of the Bianchi id…

The study proves residual finiteness for certain lattice extensions and negatively curved projective varieties.

Existence and rigidity results for lifts in Carnot groups.

We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form ω(ω-invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form ω, up to a linear isomorphism (ω-equivariant diffeomorphisms). Every abelian extensio…

The central extension of the mapping class groups of punctured surfaces of finite type that arises in quantum Teichmüller theory is 12 times the Meyer class plus the Euler classes of the punctures. This is analogous to the result obtained in \cite{FS} for the Thompson groups.

We describe a finite presentation of $\mathcal{T}_{g,r}$ for $g \geq 3$. % or $(g,r)=(2,0)$. Here $\mathcal{T}_{g,r}$ is the universal central extension of the mapping class group of the surface of genus $g$ with $r$-boundaries. We also investigate the case $g=2$,

Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration proce…

We classify the Markov traces factoring through the Birman-Wenzl-Murakami (BMW) algebras. For this purpose, we define a common `cover' for the two variations of the BMW-algebra originating from the quantum orthogonal/symplectic duality, which are responsible for the so-called `Dubrovnik' variation of the Kauffman polyn…

Action stabilizing bundle gerbe leads to Lie group extension.

In this paper, we consider the concept of connection cochain of central extensions introduced by Moriyoshi and apply it to the abelian case. We will show the relationship between connection cochain and connection $1$-form of a principal bundle whose structure group is abelian.

We briefly indicate some implications of [1] for the second Lie algebra cohomology of equivariant map algebras and (twisted multi) loop algebras.

We consider exact sequences and lower central series of surface braid groups and we explain how they can prove to be useful for obtaining representations for surface braid groups. In particular, using a completely algebraic framework, we describe the notion of extension of a representation introduced and studied recent…

We present an equivariant extension of the Thom form with respect to a vector field action, in the framework of the Mathai-Quillen formalism. The associated Topological Quantum Field Theories correspond to twisted $N=2$ supersymmetric theories with a central charge. We analyze in detail two different cases: topological…