Representations of Quantum Coordinate Algebras at Generic $q$ and Wiring Diagrams

TL;DR

This paper advances the understanding of quantum coordinate algebras at generic q by describing primitive spectra, constructing simple modules, and using wiring diagrams to connect representation theory with combinatorics.

Contribution

It generalizes previous results on tensor modules, describes primitive spectra for double Bruhat cells, and introduces wiring diagrams as a tool for quantum representation analysis.

Findings

Describes primitive spectra for quantum coordinate algebras.

Constructs bundles of simple modules over primitive spectra.

Provides a quantum Lindström's lemma for combinatorial analysis.

Abstract

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 Lindstr\"{o}m's lemma, which plays an important role in transforming representation problems into combinatorial ones of wiring diagrams.