M. Domokos, T. H. Lenagan
The general linear group acts on $m$-tuples of $N\times N$ matrices by simultaneous conjugation. Quantum deformations of the corresponding rings of invariants and the so-called trace rings are investigated.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Degenerate twisted traces on quantized Kleinian singularities of type A
本文研究型A量化克莱因奇点上的退化扭曲迹，揭示其维度与自同构及量子化参数的关系。
This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety V consists of the classical coordinate ring O(V) equipped with an associated Poisson structure. Conjectured relationships between primitive ideals of a generic quantized coordinate ring A and symplectic leaves in V (relative to a semiclassical limit Poisson structure on O(V)) are discussed, as are bre…
We prove that the generic quantized coordinate ring $\mathcal{O}_q(G)$ is Auslander-regular, Cohen-Macaulay, and catenary for every connected semisimple Lie group $G$. This answers questions raised by Brown, Lenagan, and the first author. We also prove that under certain hypotheses concerning the existence of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and Cohen-Macaulay. This provides a new set of positive cases for a question of Brown and the first author.
We compute the structure of the cohomology ring for the quantized enveloping algebra (quantum group) $U_q$ associated to a finite-dimensional simple complex Lie algebra $\mathfrak{g}$. We show that the cohomology ring is generated as an exterior algebra by homogeneous elements in the same odd degrees as generate the cohomology ring for the Lie algebra $\mathfrak{g}$. Partial results are also obtained for the cohomology rings of the non-restricted quantum groups obtained from $U_q$ by specializing the parameter $q$ to a non-zero value $ε\in \mathbb{C}$.
This paper contains a survey of some ring-theoretic aspects of quantized coordinate rings, with primary focus on the prime and primitive spectra. For these algebras, the overall structure of the prime spectrum is governed by a partition into strata determined by the action of a suitable group of automorphisms of the algebra. We discuss this stratification in detail, as well as its use in determining the primitive spectrum -- under suitable conditions, the primitive ideals are precisely those prime ideals which are maximal within their strata. The discussion then turns to the global structure o…
Generators and relations are given for the subalgebra of cocommutative elements in the quantized coordinate rings of the classical groups, where the deformation parameter q is transcendental. This is a ring theoretic formulation of the well known fact that the representation theory of the quantized group is completely analogous to its classical counterpart. The subalgebras of cocommutative elements in the corresponding FRT-bialgebras (defined by Faddeev, Reshetikhin, and Takhtadzhyan) are explicitly determined, using a bialgebra embedding of the FRT-bialgebra into the tensor product of the qua…
This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common properties, there must be a common basis on which to treat them. The first part of the paper is expository. We survey a number of classes of quantized coordinate rings, as well as some related algebras that share common properties, and we record some of the basic properties known to occur for many of these algebras, culminating in stratifications of the prime sp…
The ring of differential operators on a quantized flag manifold
本文研究了量子旗流形上微分算子环的性质，并利用量子Weyl代数给出了其在仿射开子集上局部化的显式描述。
Recently, Kuniba, Okado and Yamada proved that the transition matrix of PBW-type bases of the positive-half of a quantized universal enveloping algebra $U_q(\mathfrak{g})$ coincides with a matrix coefficients of the intertwiner between certain irreducible modules over the corresponding quantized coordinate ring $A_q(\mathfrak{g})$, introduced by Soibelman. In the present article, we give a new proof of their result, by using representation theory of the $q$-boson algebra, and the Drinfeld paring of $U_q(\mathfrak{g})$.
We define two subalgebras which can be seen as the quantization of the coordinate rings of the unipotent radical of the standard positive (respectively negative) Borel subgroup of $SL_{n+1}$. We give a presentation for these algebras and show they are isomorphic. Moreover, we show that these algebras are the conivariants of a natural $O_q(T)$-coaction on $O_q(B^\pm)$. Finally, using a dual pairing, we show that these algebras are isomorphic to the quantum nilpotent Lie subalgebra of $U_q(\mathfrak{sl}_{n+1})$.
We describe the Poisson ideals and attached symplectic geometry of a cluster algebra with compatible Poisson structure. We apply these results to determine the spectrum of a quantum cluster algebra. As an application, we describe the topology on the spectra of quantized coordinate rings such as quantum matrices and the quantized function algebra of the general linear group.
Generalized Quantum Minors Generate Quantized Coordinate Rings
本文解决了F4型单连通单复代数群对应的量子坐标环𝒪_q(F4)的量子簇代数结构问题，通过统一论证证明其由广义量子子式生成，完善了该领域的相关结论。
In this paper we investigate a multi-parameter deformation $\mathfrak{B}_{r,s}^n(a,λ,δ)$ of the walled Brauer algebra which was previously introduced by Leduc (\cite{leduc}). We construct an integral basis of $\mathfrak{B}_{r,s}^n(a,λ,δ)$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $\mathfrak{B}_{r,s}^n(q)= \mathfrak{B}_{r,s}^n(q^{-1}-q,q^n,[n]_q)$ on mixed tensor space and prove that the kernel is free over the ground ring $R$ of rank independent of $R$. As an application, we prove one side of Schur--Weyl duality for…
The Poisson centralizer of the trace element is determined in the coordinate ring of SL_n endowed with the Poisson structure obtained as the semiclassical limit of its quantized coordinate ring. It turns out that this maximal Poisson-commutative subalgebra coincides with the subalgebra of invariants with respect to the adjoint action.
Integral cluster structures on quantized coordinate rings
本文研究了复简单代数群$G$的量子坐标环的积分聚类结构，证明其在不同环上具有量子聚类代数结构，并展示了经典结果在不同环上的适用性。
Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group $U_q(\mathfrak g)$ are constructed in \cite{LZ:2009} and one can use them to define some ideals of the modified quantized enveloping algebra. It is shown that the quotient algebras inherit canonical bases from the modified quantized enveloping algebra and are dual to the quantum coordinate ring defined by Kashiwara for symmetrizable Kac-Moody algebra $\mathfrak g$.
Explicit generating sets are found for all primitive ideals in the generic quantized coordinate rings of the 3x3 special and general linear groups over an arbitrary algebraically closed field. (Previously, generators were only known up to certain localizations.) The generating sets form polynormal regular sequences, from which it follows that all primitive factor algebras of these quantized coordinate rings are Auslander-Gorenstein and Cohen-Macaulay.
Quantized Geodesic Lengths for Teichmüller Spaces: Algebraic Aspects
本文基于Chekhov-Fock理论，研究量化测地长度算子的代数结构，通过量子迹映射构造量化迹-单胞映射，并验证Teschner的递推关系及不相交循环的量子迹-单胞的强交换性。
We quantize the coordinate ring of the moduli space of B-bundles on the elliptic curve. Here B is a Borel subgroup of some semisimple Lie group. We construct some representations of these algebras and study intertwining operators for these representations. We apply our constructions to produce some objects: the elliptic Belavin R-matrix, the quantization of the algebra of functions on the Grassmannian, some generalized elliptic R-matrix. We consider also the affine case and write down the explicit formula for commuting elements.
Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A
本文研究了A_{n-1}型Kleinian奇点量子化代数上的扭曲迹，给出了显式积分公式以判定正定埃尔米特形式，从而分类了幺正短星积，证实了n≤4时的唯一性猜想，并推广了离散Painlevé系统。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。