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Keywords:
quantum coordinate ring; 2nd-stage quantized cluster algebra; compatible Poisson structure
quantum coordinate ring; 2nd-stage quantized cluster algebra; compatible Poisson structure
Summary:
Let $\mathbb {C}_q[SL(n+1)]$ denote the quantum coordinate ring of the special linear group which has a quantum cluster structure denoted by $\mathcal {A}_q$. Let $\mathcal {A}_{p,q}$ be the \hbox {2nd-stage} quantization of the quantum cluster algebra $\mathcal {A}_q$, equipped with a compatible Poisson structure $\{-,-\}$. The purpose of this paper is to describe the structure of the 2nd-stage quantized cluster algebra $\mathcal {A}_{p,q}$. We prove that for $n\geq 2$, the 2nd-stage quantized cluster algebra $\mathcal {A}_{p,q}$ is trivial. Additionally, we provide a detailed description of the compatible Poisson structure on $\mathcal {A}_q$.
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