Related papers: On some cocycles which represent the Dixmier-Douad…

We exhibit a cocycle in the simplicial de Rham complex which represents the Euler class. As an application, we construct a Lie algebra cocycle on $L\mathfrak{so}(4)$.

On the basis of A. L. Carey, D. Crowley, M. K. Murray's work, we exhibit a cocycle in the simplicial de Rham complex which represents the Dixmier-Douady class.

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.

On the basis of Dupont's work, we exhibit a cocycle in the simplicial de Rham complex which represents the Chern character. We also prove the related conjecture due to Brylinski. This gives a way to construct a cocycle in a local truncated…

Given a locally compact abelian group $G$, we give an explicit formula for the Dixmier--Douady invariant of the $C^*$-algebra of the groupoid extension associated to a \v{C}ech $2$-cocycle in the sheaf of germs of continuous $G$-valued…

It was observed recently that for a fixed finite group $G$, the set of all Drinfeld centres of $G$ twisted by 3-cocycles form a group, the so-called group of modular extensions (of the representation category of $G$), which is isomorphic to…

Given a compact semisimple Lie group $G$ of rank $r$, and a parameter $q>0$, we can define new associativity morphisms in Rep(Gq) using a 3-cocycle $\Phi$ on the dual of the center of G, thus getting a new tensor category Rep(Gq)$^\Phi$.…

In this paper we generalize some of these results for loop algebras and groups as well as for the Virasoro algebra to the two-dimensional case. We define and study a class of infinite dimensional complex Lie groups which are central…

Let $M$ be a manifold with a closed, integral $(k+1)$-form $\omega$, and let $G$ be a Fr\'echet-Lie group acting on $(M,\omega)$. As a generalization of the Kostant-Souriau extension for symplectic manifolds, we consider a canonical class…

We show that when a simplicial Lie group acts on a simplicial manifold $\{X_*\}$, we can construct a bisimplicial manifold and the de Rham complex on it. This complex is quasi-isomorphic to the equivariant simplicial de Rham complex on…

We consider the group $\mathcal G$ which is the semidirect product of the group of analytic functions with values in ${\mathbb C}^*$ on the circle and the group of analytic diffeomorphisms of the circle that preserve the orientation. Then…

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…

Using groupoid $S^1$-central extensions, we present, for a compact simple Lie group $G$, an infinite dimensional model of $S^1$-gerbe over the differential stack $G/G$ whose Dixmier-Douady class corresponds to the canonical generator of the…

We exhibit an explicit construction for the second cohomology group $H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$. We show how the elements of $H^2(L, A)$ correspond one-to-one to the equivalence classes of central extensions…

The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras su(p,q) of the Cartan series A_l and the pseudo-unitary algebras u(p,q), are…

A systematic consideration of the problem of the reduction and extension of the structure group of a principal bundle is made and a variety of techniques in each case are explored and related to one another. We apply these to the study of…

We present the derivation of the 6-dimensional Eulerian Lie group of the form SO(3,C). We describe our derivation process, which involves the creation of a finite group by using permutation matrices, and the exponentiation of the adjoint…

For a compact and connected Lie group $G$, we present an explicit construction of an $\mathbb{S}^1$-gerbe over the differentiable stack $[G/G]$ in the framework of $\mathbb{S}^1$-central extensions of Lie groupoids. This gives a complete…

We give a constructive account of the fundamental ingredients of Poisson Lie theory as the basis for a description of the classical double group $D$. The double of a group $G$ has a pointwise decomposition $D\sim G\times G^*$, where $G$ and…

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…