Quantized coordinate ring

introduction

• 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

dual of quantized enveloping algebras

• $\mathfrak{g}$ : simple Lie algebra over $\mathbb{C}$
• $U_q:=U_q(\mathfrak{g})$ : quantum enveloping algebra
• $U_q^{*}=\operatorname{Hom}_{\mathbb{Q}(q)}(U_q,\mathbb{Q}(q))$
• $A_q(\mathfrak{g}):=\{\varphi \in U_q^{*} | \dim U_q \varphi, \dim \varphi U_q <\infty \}$
• We call $A_q(\mathfrak{g})$ the quantized coordinate ring