De Rham Complex for Quantized Irreducible Flag Manifolds

An analogue of Weyl's law for quantized irreducible generalized flag manifolds

We prove an analogue of Weyl's law for quantized irreducible generalized flag manifolds. By this we mean defining a zeta function, similarly to the classical setting, and showing that it satisfies the following two properties: as a functional on the quantized algebra it is proportional to the Haar state; its first singularity coincides with the classical dimension. The relevant formulae are given for the more general case of compact quantum groups.

量化旗 manifold 和量子群在根为 unity 时的非受限模块

Quantized flag manifolds and non-restricted modules over quantum groups at roots of unity

本文证明了量子群在根为 unity 时非受限模块的乘积公式，适用于特定条件的奇素数幂。

Dirac Operators on Quantum Flag Manifolds

A Dirac operator D on quantized irreducible generalized flag manifolds is defined. This yields a Hilbert space realization of the covariant first-order differential calculi constructed by I. Heckenberger and S. Kolb. All differentials df=i[D,f] are bounded operators. In the simplest case of Podles' quantum sphere one obtains the spectral triple found by L. Dabrowski and A. Sitarz.

The quantum orbit method for generalized flag manifolds

Generalized flag manifolds endowed with the Bruhat-Poisson bracket are compact Poisson homogeneous spaces, whose decompositions in symplectic leaves coincide with their stratifications in Schubert cells. In this note it is proved that the irreducible *-representations of the corresponding quantized flag manifolds are also parametrized by their Schubert cells. An important step is the determination of suitable algebraic generators of the quantized flag manifolds. These algebraic generators can be naturally expressed in terms of quantum Plucker coordinates. This note complements the paper of the…

Quantum flag manifolds as quotients of degenerate quantized universal enveloping algebras

Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the defining relations for $U_q(\mathfrak{g})$ can be slightly modified in such a way that the resulting algebra $U_q(\mathfrak{g};\mathcal{S})$ allows a homomorphism onto (an extension of) the algebra $\mathrm{Pol}(\mathbb{G}_q/\mathbb{K}_{\mathcal{S},q})$ of functions on the quantum flag manifold $\mathbb{G}_q/\mathbb{K}_{\mathcal{S},q}$ corresponding to $\mathcal…

Quantum flag varieties, equivariant quantum D-modules and localization of quantum groups

Let $\Oq(G)$ be the algebra of quantized functions on an algebraic group $G$ and $\Oq(B)$ its quotient algebra corresponding to a Borel subgroup $B$ of $G$. We define the category of sheaves on the "quantum flag variety of $G$" to be the $\Oq(B)$-equivariant $\Oq(G)$-modules and proves that this is a proj-category. We construct a category of equivariant quantum $\mathcal{D}$-modules on this quantized flag variety and prove the Beilinson-Bernsteins localization theorem for this category in the case when $q$ is not a root of unity.

Examples of quantum cluster algebras associated to partial flag varieties

We give several explicit examples of quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky, on quantized coordinate rings of partial flag varieties and their associated unipotent radicals. These structures are shown to be quantizations of the cluster algebra structures found on the corresponding classical objects by Geiss, Leclerc and Schroer, whose work generalizes that of several other authors. We also exhibit quantum cluster algebra structures on the quantized enveloping algebras of the Lie algebras of the unipotent radicals.

On Fock Space Representations of quantized Enveloping Algebras related to Non-Commutative Differential Geometry

In this paper we construct explicitly natural (from the geometrical point of view) Fock space representations (contragradient Verma modules) of the quantized enveloping algebras. In order to do so, we start from the Gauss decomposition of the quantum group and introduce the differential operators on the corresponding $q$-deformed flag manifold (asuumed as a left comodule for the quantum group) by a projection to it of the right action of the quantized enveloping algebra on the quantum group. Finally, we express the representatives of the elements of the quantized enveloping algebra correspondi…

Quantum flag manifolds, quantum symmetric spaces and their associated universal K-matrices

Let $U$ be a connected, simply connected compact Lie group with complexification $G$. Let $\mathfrak{u}$ and $\mathfrak{g}$ be the associated Lie algebras. Let $Γ$ be the Dynkin diagram of $\mathfrak{g}$ with underlying set $I$, and let $U_q(\mathfrak{u})$ be the associated quantized universal enveloping $*$-algebra of $\mathfrak{u}$ for some $0<q$ distinct from $1$. Let $\mathcal{O}_q(U)$ be the coquasitriangular quantized function Hopf $*$-algebra of $U$, whose Drinfeld double $\mathcal{O}_q(G_{\mathbb{R}})$ we view as the quantized function $*$-algebra of $G$ considered as a real algebraic …

A multiplicative property of quantum flag minors II

Let U^+ be the plus part of the quantized enveloping algebra of a simple Lie algebra and let B^* be the dual canonical basis of U^+. Let b,b' be in B* and suppose that one of the two elements is a q-commuting product of quantum flag minors. We show that b and b' are multiplicative if and only if they q-commute.

The Beilinson-Bernstein correspondence for quantized enveloping algebras

Theory of the quantized flag manifold as a quasi-scheme (non-commutative scheme) has been developed by Lunts-Rosenberg. They have formulated an analogue of the Beilinson-Bernstein correspondence using the $q$-differential operators introduced in their earlier paper. In this paper we shall establish its modified version using a class of $q$-differential operators, which is (possibly) smaller than the one used by Lunts-Rosenberg.

Spectral gaps for twisted Dolbeault-Dirac operators over the irreducible quantum flag manifolds

We show that tensoring the Laplace and Dolbeault-Dirac operators of a Kähler structure (with closed integral) by a negative Hermitian holomorphic module, produces operators with spectral gaps around zero. The proof is based on the recently established Akizuki-Nakano identity of a noncommutative Kähler structure. This general framework is then applied to the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds, and it is shown that twisting their Dirac and Laplace operators by negative line bundles produces a spectral gap, for q sufficiently close to 1. The main technical challen…

不可约量子旗流形上的 Levi-Civita 联络

Levi-Civita connection on the irreducible quantum flag manifolds

本文对一类量子齐次空间上的协变度量进行分类，证明了在不可约量子旗流形的 Heckenberger--Kolb 微积分上存在唯一的量子对称协变度量，并证明了任意实协变度量的 Levi-Civita 联络存在且唯一，推广了量子射影空间的结果。

Quantum Algebras Associated to Irreducible Generalized Flag Manifolds

The first part of this thesis deals with certain properties of the quantum symmetric and exterior algebras of Type 1 representations of $U_q(g)$ defined by Berenstein and Zwicknagl. We define a notion of a commutative algebra object in a coboundary category, and we prove that the quantum symmetric algebra of a module is the universal commutative algebra generated by that module. That is, the functor assigning to a module its quantum symmetric algebra is left adjoint to a forgetful functor. We also prove a conjecture of Berenstein and Zwicknagl, stating that the quantum symmetric and exterior c…

正nilradical的等变量子化与协变微分微分计算

Equivariant quantizations of the positive nilradical and covariant differential calculi

本文研究了正nilradical的等变量子化，并利用其构造量子旗流形上的协变微分计算。