标题： 张量积，量子 toroidal 代数的$q$- 特征和$R$- 矩阵 显示英文标题

标题： Tensor products, $q$-characters and $R$-matrices for quantum toroidal algebras

摘要： 我们引入了一种新的拓扑余积$\Delta^{\psi}_{u}$，用于所有未扭曲类型的量子环面代数$U_{q}(\mathfrak{g}_{\mathrm{tor}})$，从而在可积表示范畴$\widehat{\mathcal{O}}_{\mathrm{int}}$上定义了一个良好的张量积。这通过用一种反自同构$\psi$来扭曲 Drinfeld 余积$\Delta_{u}$定义，该反自同构交换了$U_{q}(\mathfrak{g}_{\mathrm{tor}})$的水平和垂直量子仿射子代数。 $\psi$ 的其他应用包括将著名的米基自同构从类型 $A$ 推广，并研究 $SL_{2}(\mathbb{Z})$ 的万有覆盖的一个作用。 接下来，我们研究由此产生的 $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ 张量表示，并证明了查里-普雷斯利关于仿射层次上的一系列重要结果的量子环面类似物。 特别是，它与德林费尔德多项式兼容，并且不可约因子的乘积通常仍是不可约的。 此外，我们还证明了张量积的 $q$-特征等于其因子的 $q$-特征的乘积。 此外，我们得到了带谱参数的$R$-矩阵，它为（三角、量子）Yang-Baxter方程提供了解，并使$\widehat{\mathcal{O}}_{\mathrm{int}}$具有亚纯辫子结构。 这些进一步为每个模给出了一个交换的转移矩阵族。 显示英文摘要

摘要： We introduce a new topological coproduct $\Delta^{\psi}_{u}$ for quantum toroidal algebras $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ in all untwisted types, leading to a well-defined tensor product on the category $\widehat{\mathcal{O}}_{\mathrm{int}}$ of integrable representations. This is defined by twisting the Drinfeld coproduct $\Delta_{u}$ with an anti-involution $\psi$ of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ that swaps its horizontal and vertical quantum affine subalgebras. Other applications of $\psi$ include generalising the celebrated Miki automorphism from type $A$, and an action of the universal cover of $SL_{2}(\mathbb{Z})$. Next, we investigate the ensuing tensor representations of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$, and prove quantum toroidal analogues for a series of influential results by Chari-Pressley on the affine level. In particular, there is a compatibility with Drinfeld polynomials, and the product of irreducibles is generically irreducible. We moreover show that the $q$-character of a tensor product is equal to the product of $q$-characters for its factors. Furthermore, we obtain $R$-matrices with spectral parameter which provide solutions to the (trigonometric, quantum) Yang-Baxter equation, and endow $\widehat{\mathcal{O}}_{\mathrm{int}}$ with a meromorphic braiding. These moreover give rise to a commuting family of transfer matrices for each module.