$\Ughr$ invariant quantization on some homogeneous manifolds

Equivariant quantization on quotients of simple Lie groups by reductive subgroups of maximal rank

We consider a class of homogeneous manifolds including all semisimple coadjoint orbits. We describe manifolds of that class admitting deformation q uantizations equivariant under the action of $G$ and the corresponding quantum group. We also classify Poisson brackets relating to such quantizations.

A global quantum duality principle for subgroups and homogeneous spaces

For a complex or real algebraic group G, with g:=Lie(G), quantizations of global type are suitable Hopf algebras F_q[G] or U_q(g) over C[q,q^{-1}]. Any such quantization yields a structure of Poisson group on G, and one of Lie bialgebra on g : correspondingly, one has dual Poisson groups G^* and a dual Lie bialgebra g^*. In this context, we introduce suitable notions of quantum subgroup and of quantum homogeneous space, in three versions: weak, proper and strict (also called "flat" in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneo…

Quantization of classical dynamical $r$-matrices with nonabelian base

We construct some classes of dynamical $r$-matrices over a nonabelian base, and quantize some of them by constructing dynamical (pseudo)twists in the sense of Xu. This way, we obtain quantizations of $r$-matrices obtained in earlier work of the second author with Schiffmann and Varchenko. A part of our construction may be viewed as a generalization of the Donin-Mudrov nonabelian fusion construction. We apply these results to the construction of equivariant star-products on Poisson homogeneous spaces, which include some homogeneous spaces introduced by De Concini.

齐次星积的分类与约化

Classification and Reduction of Homogeneous Star Products

本文利用第二李代数胚上同调对李代数胚对偶上的齐次星积进行分类，并将其推广到可投影星积，证明了在该框架下量子化与约化可交换。

Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

Let $\mathfrak g$ be a finite dimensional split semisimple Lie algebra and $λ$ a weight of $\mathfrak g$. Let $F$ be the algebra of quantized regular functions on the connected simply connected group $G$ corresponding to $\mathfrak g$. In the present paper we introduce a certain subspace $F'$ of $F$ (which is not necessary a subalgebra of $F$) and endow it with an associative $\star$-product using the so-called reduced fusion element. We prove that the algebra $(F',\star)$ is isomorphic to $(L(λ))_{fin}$, where $L(λ)$ is the irreducible highest weight $\check{U}_q\mathfrak g$-module and "$fin$…

Quantum planes and quantum cylinders from Poisson homogeneous spaces

Quantum planes and a new quantum cylinder are obtained as quantization of Poisson homogeneous spaces of two different Poisson structures on classical Euclidean group E(2).

A quantum duality principle for coisotropic subgroups and Poisson quotients

We develop a quantum duality principle for coisotropic subgroups of a (formal) Poisson group and its dual. Namely, starting from a quantum coisotropic subgroup (for a quantization of a given Poisson group) we provide functorial recipes to produce quantizations of the dual coisotropic subgroup (in the dual formal Poisson group). By the natural link between subgroups and homogeneous spaces, we argue a quantum duality principle for Poisson homogeneous spaces which are Poisson quotients, i.e. have at least one zero-dimensional symplectic leaf. As an application, we provide an explicit quantization…

Algebraic Framework for Quantization of Nonultralocal Models

Extension of the braid relations to the multiple braided tensor product of algebras that can be used for quantization of nonultralocal models is presented. The Yang--Baxter--type consistency conditions as well as conditions for the existence of the multiple coproduct (monodromy matrix), which can be used for construction of the commuting subalgebra, are given. Homogeneous and local algebras are introduced, and simplification of the Yang--Baxter--type conditions for them is shown. The Yang--Baxter--type equations and multiple coproduct conditions for homogeneous and local of order 2 algebras ar…

Quantization of Poisson manifolds from the integrability of the modular function

We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows very singular polarizations. In particular we consider the case when the modular function is "multiplicatively integrable", i.e. when the space of leaves of the polarization inherits a groupoid structure. If suitable regularity conditions are satisfied, then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying t…

On quantizations of complex contact manifolds

A (holomorphic) quantization of a complex contact manifold is a filtered algebroid stack which is locally equivalent to the ring E of microdifferential operators and which has trivial graded. The existence of a canonical quantization has been proved by Kashiwara. In this paper we consider the classification problem, showing that the above quantizations are classified by the first cohomology group with values in a certain sheaf of homogeneous forms. Secondly, we consider the problem of existence and classification for quantizations given by algebras.

Quantization of a Poisson structure on products of principal affine spaces

We give the analogue for Hopf algebras of the polyuble Lie bialgebra construction by Fock and Rosli. By applying this construction to the Drinfeld-Jimbo quantum group, we obtain a deformation quantization $\mathbb{C}_\hslash[(N \backslash G)^m]$ of a Poisson structure $π^{(m)}$ on products $(N \backslash G)^m$ of principal affine spaces of a connected and simply connected complex semisimple Lie group $G$. The Poisson structure $π^{(m)}$ descends to a Poisson structure $π_m$ on products $(B \backslash G)^m$ of the flag variety of $G$ which was introduced and studied by the Lu and the author. An…

On the Fedosov Deformation Quantization beyond the Regular Poisson Manifolds

A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang-Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF's is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson brackets on two-plane.

Quantization of the Nonlinear Sigma Model Revisited

We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization. The symmetries we consider are twofold: (i) diffeomorphism covariance for a general target manifold; (ii) a transitive group of isometries when the target manifold is a homogeneous space. We show that there are no anomalies in case (i) and that (ii) is also anomaly-free under additional…

Berezin quantization of noncommutative projective varieties

We use operator algebras and operator theory to obtain new result concerning Berezin quantization of compact Kähler manifolds. Our main tool is the notion of subproduct systems of finite-dimensional Hilbert spaces, which enables all involved objects, such as the Toeplitz operators, to be very conveniently expressed in terms of shift operators compressed to a subspace of full Fock space. This subspace is not required to be contained in the symmetric Fock space, so from finite-dimensional matrix algebras we can construct noncommutative manifolds with extra structure generalizing that of a projec…

An explicit formula for PBW quantization

Let $k$ be a field of characteristic zero, $\g$ a $k$-Lie algebra, $e:S\g@>>>U\g$ the symmetrization map. The PBW quantization is the one parameter family of associative products: $$ x\star_t y=\sum_{p=0}^\infty B_p(x,y)t^p\qquad (t\in k) $$ where $B_p$ is the homogeneous component of degree $-p$ of the map $B:S\g\otimes_kS\g@>>>S\g$, $B(x,y)=e^{-1}(exey)$. In this paper we give an explicit formula for $B$. As an application, we prove that for each $p\ge 0$, $B_p$ is a bidifferential operator of order $\le p$.

A quantum duality principle for subgroups and homogeneous spaces

We develop a quantum duality principle for subgroups of a Poisson group and its dual, in two formulations. Namely, in the first one we provide functorial recipes to produce quantum coisotropic subgroups in the dual Poisson group out of any quantum subgroup (in a tautological sense) of the initial Poisson group, while in the second one similar recipes are given only starting from coisotropic subgroups. In both cases this yields a Galois-type correspondence, where a quantum coisotropic subgroup is mapped to its complementary dual; moreover, in the first formulation quantum coisotropic subgroups …

Quantization by cochain twists and nonassociative differentials

We show that several standard associative quantizations in mathematical physics can be expressed as cochain module-algebra twists in the spirit of Moyal products at least to $O(\hbar^3)$, but to achieve this we twist not by a 2-cocycle but by a 2-cochain. This implies a hidden nonassociavitity not visible in the algebra itself but present in its deeper noncommutative differential geometry, a phenomenon first seen in our previous work on semiclassicalisation of differential structures. The quantisations are induced by a classical group covariance and include: enveloping algebras $U_\hbar(g)$ as…

Noncommutative solitons on Kahler manifolds

We construct a new class of scalar noncommutative multi-solitons on an arbitrary Kahler manifold by using Berezin's geometric approach to quantization and its generalization to deformation quantization. We analyze the stability condition which arises from the leading 1/hbar correction to the soliton energy and for homogeneous Kahler manifolds obtain that the stable solitons are given in terms of generalized coherent states. We apply this general formalism to a number of examples, which include the sphere, hyperbolic plane, torus and general symmetric bounded domains. As a general feature we no…

七球面量子化

Seven Sphere Quantization

本文将标准切触七球面作为四元数酉群的齐性空间，通过推广 Holstein–Primakoff 机制构造 Fedosov 形式联络的切触类比，并证明其收敛性自然过滤辛旋量丛，从而得到四元数酉群的有限维酉不可约表示。

关于 Yangian 的唯一性

On the uniqueness of Yangians

本文证明了单李代数多项式当前代数的齐次量子化至多唯一的断言，结合上同调与计算方法，输出了此类量子化的生成元与约化关系展示。

具有泊松稳定子的齐性空间上的量化函数代数

Quantized algebras of functions on homogeneous spaces with Poisson stabilizers

本文研究带泊松稳定子的齐性空间G/K上函数代数的量化C(G_q/K_q)，分类其不可约表示、给出合成列，刻画其谱拓扑，证明该C*-代数族构成连续场并实现严格形变量化，且与C(G/K)典范KK等价。

Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic $\mathfrak{o}^{\star}(4)$ symmetries

We construct firstly the complete list of five quantum deformations of $D=4$ complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})\cong \mathfrak{o}(3;\mathbb{C})\oplus \mathfrak{o}(3;\mathbb{C})$, describing quantum rotational symmetry of four-dimensional complex space-time, in particular we provide the corresponding universal quantum $R$-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\ma…