论文标题
量子坐标为代数的表示，在通用$ Q $和接线图
Representations of Quantum Coordinate Algebras at Generic $q$ and Wiring Diagrams
论文作者
论文摘要
本文专用于量子坐标的代表理论代数$ \ mathbb {c} _q [g] $，用于半imple lie group $ g $和一个通用参数$ q $。通过检查普通元素在张量模块上的作用，我们将Levendorski和Soibelman的结果推广为[22]中的最高权重模块。对于双重bruhat cell $ g^{w_1，w_2} $，我们描述了原始光谱$ \ mathrm {prim} \，\ mathbb {c} _q [g] _ {w_1，w_1，w_2} $，并以一种新的方式构建$（W_1，w_1，type Modes $（w_1，w_1，type） $ \ mathrm {Prim} \，\ Mathbb {c} _q [g] _ {w_1，w_2} $，提供$ \ Mathrm {Supp}（w_1）\ cap \ cap \ mathrm {supprm {supp}（w_2）= \ varnothing $或足够的pivotemement。捆绑包的纤维被证明是二维量子圆环$ L_Q（2）$的简单模块光谱的产物。作为我们理论的应用，我们推断出张量模块简单的等效条件，并在$ g = sl_3（\ mathbb {c}）$时为每个原始理想构造一些简单的模块。这完成了$ \ Mathbb {C} _Q [SL_3] $的Dixmier程序。 Fomin和Zelevinsky在其总阳性研究中引入的接线图（参见[3,9]）是计算A型情况中广义量子未成年人对张量模块的作用的主要工具。我们获得了Lindström的引理的量子版本，该版本在将表示问题转换为接线图的组合问题中起着重要作用。
This paper is devoted to the representation theory of quantum coordinate algebra $\mathbb{C}_q[G]$, for a semisimple Lie group $G$ and a generic parameter $q$. By inspecting the actions of normal elements on tensor modules, we generalize a result of Levendorski and Soibelman in [22] for highest weight modules. For a double Bruhat cell $G^{w_1,w_2}$, we describe the primitive spectra $\mathrm{prim}\,\mathbb{C}_q[G]_{w_1,w_2}$ in a new fashion, and construct a bundle of $(w_1,w_2)$ type simple modules onto $\mathrm{prim}\,\mathbb{C}_q[G]_{w_1,w_2}$, provided $\mathrm{Supp}(w_1)\cap\mathrm{Supp}(w_2)=\varnothing$ or enough pivot elements. The fibers of the bundle are shown to be products of the spectrums of simple modules of 2-dimensional quantum torus $L_q(2)$. As an application of our theory, we deduce an equivalent condition for the tensor module to be simple, and construct some simple modules for each primitive ideal when $G=SL_3(\mathbb{C})$. This completes the Dixmier's program for $\mathbb{C}_q[SL_3]$. The wiring diagrams, introduced by Fomin and Zelevinsky in their study of total positivity (cf. [3,9]), is the main tool to compute the action of generalized quantum minors on tensor modules in the type A case. We obtain a quantum version of Lindström's lemma, which plays an important role in transforming representation problems into combinatorial ones of wiring diagrams.
