Quantized coordinate ring

introduction

• $\mathfrak{g}$ : simple Lie algebra over $\mathbb{C}$
• $G$ : connected, simply-connected simple algebraic group with Lie algebra $\mathfrak{g}$

dual of quantized enveloping algebras

QEA

• $q\in \mathbb{C}^{\times}$ not a root of 1
• $U_q:=U_q(\mathfrak{g})=\langle k_i^{\pm},e_i,f_i :i\in I \rangle$ : quantum enveloping algebra

quantized coordinate algebra

• $U_q^{*}=\operatorname{Hom}_{\mathbb{Q}(q)}(U_q,\mathbb{Q}(q))$
• $A_q(\mathfrak{g}):=\{\varphi \in U_q^{*} | \dim U_q \varphi, \dim \varphi U_q <\infty \}$
  • also denoted by $\mathbb{C}_q[G]$
• We call $A_q(\mathfrak{g})$ the quantized coordinate ring

comodules and modules

• $\mathbb{C}_q[G]$-comodules = locally finite $U_q(\mathfrak{g})$-modules of type 1

thm (Soibelman)

result of Kuniba, Okado and Yamada

thm

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.

cluster theory

• Monoidal categorifications of cluster algebras
• $\mathbb{C}[N]$ is Hopf dual to $U(\mathfrak{n})$ where $\mathfrak{n}=Lie(N)$
• Ringel, Lusztig : Geometric realization of $U_q(\mathfrak{n})$ via constructible sheaves on varieties of $\mathbb{C}Q$-modules
• Lusztig : Geometric realization of $U(n)$ via constructible functions on varieties of $\Lambda$-modules
• Geiss-Leclerc-S : Dualizing Lusztig's construction, get a cluster character

related items

• Q-boson algebras

expositions

• Saito, Yoshihisa (University of Tokyo) Quantized coordinate rings, PBW bases and q-boson algebras
• Tanisaki, Toshiyuki (Osaka City University) Modules over quantized coordinate algebras and PBW-bases

articles

• Geiss, Christof, Bernard Leclerc, and Jan Schröer. “Quivers with Relations for Symmetrizable Cartan Matrices III: Convolution Algebras.” arXiv:1511.06216 [math], November 19, 2015. http://arxiv.org/abs/1511.06216.
• Geiss, Christof, Bernard Leclerc, and Jan Schröer. “Quivers with Relations for Symmetrizable Cartan Matrices II: Change of Symmetrizers.” arXiv:1511.05898 [math], November 18, 2015. http://arxiv.org/abs/1511.05898.
• Oya, Hironori. “Representations of Quantized Function Algebras and the Transition Matrices from Canonical Bases to PBW Bases.” arXiv:1501.01416 [math], January 7, 2015. http://arxiv.org/abs/1501.01416.
• Tanisaki, T. “Modules over Quantized Coordinate Algebras and PBW-Bases.” arXiv:1409.7973 [math], September 28, 2014. http://arxiv.org/abs/1409.7973.

books

• Korogodski, Leonid I., and Yan S. Soibelman. Algebras of Functions on Quantum Groups. American Mathematical Soc., 1998.