Star Products on Coadjoint Orbits

Deformation quantization and quantum coadjoint orbits of SL(2,R)

In this article we describe the coadjoint orbits of SL(2,$\mathbb R$). After choosing polarizations for each orbits, we pointed out the corresponding quantum coadjoint orbits and therefore unitary representations of SL(2,$\mathbb R$) via deformation quantization.

Twisted conjugacy classes, coadjoint orbits of loop groups and D-branes in the WZW-model

We show that untwisted respectively twisted conjugacy classes of a compact and simply connected Lie group which satisfy a certain integrality condition correspond naturally to irreducible highest weight representations of the corresponding affine Lie algebra. Along the way, review the classification of twisted conjugacy classes of a simply connected compact Lie group $G$ and give a description of their stabilizers in terms of the Dynkin diagram of the corresponding twisted affine Lie algebra.

Characters, Coadjoint Orbits and Duistermaat-Heckman Integrals

The asymptotics of characters $χ_{kλ}(\exp(h/k))$ of irreducible representations of a compact Lie group $G$ for large values of the scaling factor $k$ are given by Duistermaat-Heckman (DH) integrals over coadjoint orbits of $G$. This phenomenon generalises to coadjoint orbits of central extensions of loop groups $\widehat{LG}$ and of diffeomorphisms of the circle $\widehat{\rm Diff}(S^1)$. We show that the asymptotics of characters of integrable modules of affine Kac-Moody algebras and of the Virasoro algebra factorize into a divergent contribution of the standard form and a convergent contrib…

Comparison and Continuity of Wick-type Star Products on certain coadjoint orbits

In this paper we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev-Lachowska and Karabegov and we prove that they agree in general. In the case of the 2-sphere we establish the continuity of the star product, thereby allowing for a completion to a Fréchet algebra.

A trace formula for the quantization of coadjoint orbits

The main goal of this paper is to compute the characteristic class of the Alekseev-Lachowska *-product on coadjoint orbits. We deduce an analogue of the Weyl dimension formula in the context of deformation quantization.

Quantum coadjoint orbits of GL(n) and generalized Verma modules

In our previous paper, we constructed an explicit GL(n)-equivariant quantization of the Kirillov--Kostant-Souriau bracket on a semisimple coadjoint orbit. In the present paper, we realize that quantization as a subalgebra of endomorphisms of a generalized Verma module. As a corollary, we obtain an explicit description of the annihilators of generalized Verma modules over U(gl(n)). As an application, we construct real forms of the quantum orbits and classify finite dimensional representations. We compute the non-commutative Connes index for basic homogenous vector bundles over the quantum orbit…

Quantum Strata of Coadjoint Orbits

In this paper we construct quantum analogs of strata of coadjoint orbits and describe their representations. This kind objects play an important role in describing quantum groups as repeated extensions of quantum strata.

$\U_q(sl(n))$-invariant quantization of symmetric coadjoint orbits via reflection equation algebra

We study relations between the two-parameter $\U_q(sl(n))$-invariant deformation quantization on $sl^*(n)$ and the reflection equation algebra. The latter is described by a quantum permutation on $\End(\C^n)$ given explicitly. The reflection equation algebra is used for constructing the one-parameter quantization on coadjoint orbits, including symmetric and certain bisymmetric and nilpotent ones. Our approach is based on embedding the quantized function algebras on the orbits into the algebra of functions on the quantum group $SL_q(n)$ via reflection equation algebra characters.

Strict quantization of coadjoint orbits

We obtain a family of strict $\hat G$-invariant products on the space of holomorphic functions on a semisimple coadjoint orbit of a complex connected semisimple Lie group $\hat G$. By restriction, we also obtain strict $G$-invariant products $*_\hbar$ on a space $A(O)$ of certain analytic functions on a semisimple coadjoint orbit $O$ of a real connected semisimple Lie group $G$. The space $A(O)$ endowed with one of the products $*_\hbar$ is a Fréchet algebra, and the formal expansion of the products around $\hbar = 0$ determines a formal deformation quantization of $O$, which is of Wick type i…

$\Uh$ invariant Quantization of Coadjoint Orbits and Vector Bundles over them

Let M be a coadjoint semisimple orbit of a simple Lie group G. Let $U_h(\g)$ be a quantum group corresponding to G. We construct a universal family of $U_h(\g)$ invariant quantizations of the sheaf of functions on M and describe all such quantizations. We also describe all two parameter $U_h(\g)$ invariant quantizations on M, which can be considered as $U_h(\g)$ invariant quantizations of the Kirillov-Kostant-Souriau (KKS) Poisson bracket on M. We also consider how those quantizations relate to the natural polarizations of M with respect to the KKS bracket. Using polarizations, we quantize the…

Double quantization on coadjoint representations of simple Lie groups and its orbits

Let M be a manifold with an action of a Lie group G, $\A$ the function algebra on M. The first problem we consider is to construct a $U_h(\g)$ invariant quantization, $\A_h$, of $\A$, where $U_h(\g)$ is a quantum group corresponding to G. Let s be a G invariant Poisson bracket on M. The second problem we consider is to construct a $U_h(\g)$ invariant two parameter (double) quantization, $\A_{t,h}$, of $\A$ such that $\A_{t,0}$ is a G invariant quantization of $s$. We call $\A_{t,h}$ a $U_h(\g)$ invariant quantization of the Poisson bracket s. In the paper we study the cases when G is a simple …

Geometric quantization and quantum moment maps on coadjoint orbits and Kähler-Einstein manifolds

Deformation quantization and geometric quantization on Kähler manifolds give the mathematical description of the algebra of quantum observables and the Hilbert spaces respectively, where the later forms a representation of quantum observables asymptotically via Toeplitz operators. When there is a Hamiltonian $G$-action on a Kähler manifold, there are associated symmetries on both the quantum algebra and representation aspects. We show that in nice cases of coadjoint orbits and Kähler-Einstein manifolds, these symmetries are strictly compatible (not only asymptotically).

Invariant quantization in one and two parameters on semisimple coadjoint orbits of simple Lie groups

We study one and two parameter quantizations of the function algebra on a semisimple orbit in the coadjoint representation of a simple Lie group subject to the condition that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group. We prove that the corresponding Poisson bracket must be the sum of the so-called R-matrix bracket and an invariant bracket. We classify such brackets for all semisimple orbits and show that they form a family of dimension equal to the rank equal to the second cohomology group of the orbit and then we quantize these b…

Quantization of Equivariant Vector Bundles

The quantization of vector bundles is defined. Examples are constructed for the well controlled case of equivariant vector bundles over compact coadjoint orbits. (Coadjoint orbits are symplectic spaces with a transitive, semisimple symmetry group.) In preparation for the main result, the quantization of coadjoint orbits is discussed in detail. This subject should not be confused with the quantization of the total space of a vector bundle such as the cotangent bundle.

Quantization of Double Covers of Nilpotent Coadjoint Orbits I: Noncommutative Models

We construct by geometric methods a noncommutative model E of the algebra of regular functions on the universal (2-fold) cover M of certain nilpotent coadjoint orbits O for a complex simple Lie algebra g. Here O is the dense orbit in the cotangent bundle of the generalized flag variety X associated to a complexified Cartan decomposition g=(p^+)+k+(p^-) where p^+- are Jordan algebras by the TKK construction. We obtain E as the algebra of g-finite differential operators on a smooth Lagrangian subvariety in M where g is given by differential operators twisted according to a critical parameter. Af…

Quantization of function algebras on semisimple orbits in $\g^*$

In this paper we describe a multiparameter deformation of the function algebra of a semisimple coadjoint orbit. In the first section we use the representation of the Lie algebra on a generalized Verma module to quantize the Kirillov bracket on the family of semisimple coadjoint orbits of a given orbit type. In the second section we extend this construction to define a deformation in the category of representations of the quantized enveloping algebra. In an earlier paper we used cohomological methods to prove the existence of a two parameter family quantizing a compatible pair of Poisson bracke…

Rational version of Archimedes symplectomorphysm and birational Darboux coordinates on coadjoint orbit of $GL(N,C)$

A set of all linear transformations with a fixed Jordan structure $J$ is a symplectic manifold isomorphic to the coadjoint orbit $\mathcal O (J)$ of $GL(N,C)$. Any linear transformation may be projected along its eigenspace to (at least one) coordinate subspace of the complement dimension. The Jordan structure $\tilde J$ of the image is defined by the Jordan structure $J$ of the pre-image, consequently the projection $\mathcal O (J)\to \mathcal O (\tilde J)$ is the mapping of the symplectic manifolds. It is proved that the fiber $\mathcal E$ of the projection is a linear symplectic space and t…

通过形变量子包络代数构造的量子半单余伴随轨道的多项式族

Polynomial families of quantum semisimple coadjoint orbits via deformed quantum enveloping algebras

本研究利用形变量子包络代数构造Drinfeld-Jimbo形变表示范畴上的半单左模范畴多项式族，建立与中间Levi子代数诱导模范畴的比较定理，相关内容在Lusztig整性框架下讨论。