论文标题
量子仿射和双量子仿射代数$ \ mathfrak a_1 $上的重量限制模块
Weight-finite modules over the quantum affine and double quantum affine algebras of type $\mathfrak a_1$
论文作者
论文摘要
我们在类型上定义了$ \ mathfrak a_1 $ Quantum Aggine代数$ \ dot {\ Mathrm {u}} _ q（\ Mathfrak a_1）$和类型$ \ mathfrak a_1 $ double Quant Quant Quand Quant量Algebra a Q（\ Mathrm {u}} _ Q（\ Mathrm {u}} _ $ \ ddot {\ mathrm {u}} _ q（\ mathfrak a_1）$，我们在上一篇论文中介绍。在这两种情况下，我们都会在这些类别中对简单对象进行分类。在量子仿射的情况下，我们证明它们与简单的有限维$ \ dot {\ mathrm {u}} _ q（\ Mathfrak a_1）$ - 由Chari和Pressley归类为Chari and Pressley的最高（Rational and $ \ ell $ \ $ -Dominant）的模块。多项式。在双重量子仿射情况下，我们表明简单的重量 - 限制模块由其（$ t $ - dominant）最高$ t $ - 自大空间，这是一个subalgebra $ \ ddot {\ mathrm {u}}} _ q^0（\ mathfrak a_1）$的简单模块的家族$ \ ddot {\ mathrm {u}} _ q（\ mathfrak a_1）$，它是对椭圆形霍尔代数的拆分扩展的构象。在双量子仿射情况下，分类的证明依赖于构建量子仿射设置中出现的评估模块的双量子仿射类似物。
We define the categories of weight-finite modules over the type $\mathfrak a_1$ quantum affine algebra $\dot{\mathrm{U}}_q(\mathfrak a_1)$ and over the type $\mathfrak a_1$ double quantum affine algebra $\ddot{\mathrm{U}}_q(\mathfrak a_1)$ that we introduced in a previous paper. In both cases, we classify the simple objects in those categories. In the quantum affine case, we prove that they coincide with the simple finite-dimensional $\dot{\mathrm{U}}_q(\mathfrak a_1)$-modules which were classified by Chari and Pressley in terms of their highest (rational and $\ell$-dominant) $\ell$-weights or, equivalently, by their Drinfel'd polynomials. In the double quantum affine case, we show that simple weight-finite modules are classified by their ($t$-dominant) highest $t$-weight spaces, a family of simple modules over the subalgebra $\ddot{\mathrm{U}}_q^0(\mathfrak a_1)$ of $\ddot{\mathrm{U}}_q(\mathfrak a_1)$ which is conjecturally isomorphic to a split extension of the elliptic Hall algebra. The proof of the classification, in the double quantum affine case, relies on the construction of a double quantum affine analogue of the evaluation modules that appear in the quantum affine setting.
