Related papers: Coadjoint orbits of Lie groupoids

We show that the leaves of an LA-groupoid which pass through the unit manifold are, modulo a connectedness issue, Lie groupoids. We illustrate this phenomenon by considering the cotangent Lie algebroids of Poisson groupoids thus obtaining…

The Lie-Poisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are…

We study the geometrical structure of the coadjoint orbits of an arbitrary complex or real Lie algebra ${\mathfrak g}$ containing some ideal ${\mathfrak n}$. It is shown that any coadjoint orbit in ${\mathfrak g}^*$ is a bundle with the…

We prove that any coadjoint orbit with real eigenvalues of a complex semisimple Lie group, equipped with the real part of the canonical holomorphic symplectic form, is symplectomorphic to the cotangent bundle of a (partial) flag manifold.…

As we said in our previous work [4], the main idea of our research is to introduce a class of Lie groupoids by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid, which we called coadjoint Lie groupoids. In…

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the…

We give various realizations of the adjoint orbits of a semi-simple Lie group and describe their symplectic geometry. We then use these realizations to identify a family of Lagrangean submanifolds of the orbits.

Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation theory, geometrical quantization, theory of magnetism, quantum optics etc. As geometric objects the orbits were…

Let G be a noncompact real semisimple Lie group. The regular coadjoint orbits of G can be partitioned into a finite set of types. We show that on each regular orbit, the Iwasawa decomposition induces a left-invariant foliation which is…

Let G be an n-dimensional semisimple compact and connected Lie group acting on both the Lie algebra g of G and its dual g*. We show that a nondegenerate Killing form of G induces an Ad*-equivariant isomorphism of g onto g* which, in turn,…

We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian…

This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant K\"ahlerian structure. These…

We consider canonical symplectic structure on the moduli space of flat ${\g}$-connections on a Riemann surface of genus $g$ with $n$ marked points. For ${\g}$ being a semisimple Lie algebra we obtain an explicit efficient formula for this…

This work is motivated by a result of Drinfeld on Poisson homogeneous spaces. For each Poisson manifold $P$ with a Poisson action by a Poisson Lie group $G$, we describe a Lie algebroid structure on the direct sum vector bundle $P \times…

In this paper we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group ${\rm GL}_r(\mathcal{K}_{\mathbb{P}^1_x})$ with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at…

This note provides a formula for the character of the Lie algebra of the fundamental group of a surface, viewed as a module over the symplectic group.

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to Lie algebras of dimension 7 such that the nilradical of them is 5-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ in Table…

We answer the natural question: when are a regular Poisson structure along with a complex structure transverse to its symplectic leaves induced by generalized complex structure? The leafwise symplectic form and transverse complex structure…

In recent work (\cite{KW1},\cite{KW2}), Kostant and Wallach construct an action of a simply connected Lie group $A\simeq \mathbb{C}^{{n\choose 2}}$ on $gl(n)$ using a completely integrable system derived from the Poisson analogue of the…

Let $(V,\gamma )$ be a real finite dimensional vector space with a symmetric bilinear form $\gamma $ whose kernel is spanned by a nonzero vector. The set of invertible real linear mappings of $(V, \gamma )$ into itself forms a Lie group…