We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q,t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q,t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.

中文翻译：

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一种量子簇代数方法来表示简单的量子仿射代数

我们在简列量子仿射代数的有限维表示范畴的某个幺半群子范畴的量子格洛腾迪克环上建立了一个量子簇代数结构。此外，某些不可约表示的 (q,t) 字符，其中基本表示，是作为量子簇变量获得的。这种方法提供了一种计算这些 (q,t) 字符的新算法。作为一个应用，我们证明了量子仿射代数的 Borel 子代数表示的更大范畴的量子 Grothendieck 环，在之前的工作中定义为量子簇代数，确实包含该范畴的著名量子 Grothendieck 环有限维表示。最后，我们在一个具体的例子中展示了我们的算法。