论文标题
移动量子仿射代数的表示
Representations of shifted quantum affine algebras
论文作者
论文摘要
我们开发了移动的量子仿射代数的表示理论$ \ MATHCAL {u} _Q^μ（\ hat {\ Mathfrak {g}}）$及其截断的截断，这些截断出现在量化的量化的K主理论库仑分支3D $ n = 4 $ s susy susy Quiver Quiver Quiver Quiver eguge的研究中。我们的方法基于新技术，这些技术是在延宗主义或普通量子仿射代数的情况下是新的：从量子仿射代数$ \ Mathcal {U} _q（\ hatfrak {\ hatfrak {g Mathfrak {g}）$，TACERORS $ {OO {OO oforts and cantical的差异亚代数方面实现。 $ \ Mathcal {U} $ \ Mathcal {u} _Q（\ hat} _q（\ hat {\ hat {\ Mathfrak {g Mathfrak {g}}）$的$ \ MATHCAL {双重插值。我们首先介绍$ \ Mathcal {u} _Q^μ的表示类别$ \ Mathcal {O} _ $ $（\ hat {\ Mathfrak {g}}）$，我们对其简单对象进行了分类。然后，我们建立了融合产品的存在，并在Grothendieck组的总和$ k_0（\ Mathcal {o}_μ）$上获得了一个环结构。我们对$ \ Mathcal {u} _Q^μ的简单有限维表示（\ hat {\ Mathfrak {g}}）$进行了分类，我们在有限二维表示的Grothendieck环上获得了群集代数结构。我们证明截断只有有限数量的简单表示形式，我们在简单模块上引入了相关的部分订购。最终，我们对非简单截断的简单模块的参数化表示了一个猜想。我们有几个证据，包括简单有限维表示的一般结果。
We develop the representation theory of shifted quantum affine algebras $\mathcal{U}_q^μ(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge theories. Our approach is based on novel techniques, which are new in the cases of shifted Yangians or ordinary quantum affine algebras as well : realization in terms of asymptotical subalgebras of the quantum affine algebra $\mathcal{U}_q(\hat{\mathfrak{g}})$, induction and restriction functors to the category $\mathcal{O}$ of representations of the Borel subalgebra $\mathcal{U}_q(\hat{\mathfrak{b}})$ of $\mathcal{U}_q(\hat{\mathfrak{g}})$, relations between truncations and Baxter polynomiality in quantum integrable models, parametrization of simple modules via Langlands dual interpolation. We first introduce the category $\mathcal{O}_μ$ of representations of $\mathcal{U}_q^μ(\hat{\mathfrak{g}})$ and we classify its simple objects. Then we establish the existence of fusion products and we get a ring structure on the sum of the Grothendieck groups $K_0(\mathcal{O}_μ)$. We classify simple finite-dimensional representations of $\mathcal{U}_q^μ(\hat{\mathfrak{g}})$ and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We prove a truncation has only a finite number of simple representations and we introduce a related partial ordering on simple modules. Eventually, we state a conjecture on the parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations.
