论文标题
量子仿射代数的PBW理论
PBW theory for quantum affine algebras
论文作者
论文摘要
令$ u_q'（\ mathfrak {g}）$为任意类型的量子仿射代数，让$ \ mathcal {c} _ {\ mathfrak {g}} $为hernandez-leclerc的类别。我们可以将量子Aggine Schur-weyl双重功能函数$ f_d $与双重数据$ d $ in $ \ MATHCAL {C} _ {\ MATHFRAK {g}} $相关联。我们介绍了一个强（完整的）双重数据$ D $的概念，并证明当$ d $很强时，诱导的duality functor $ f_d $将简单的模块发送到简单的模块，并保留不变的$λ$和$λ$和$λ^\ infty $。接下来，我们定义反射$ \ MATHCAL {S} _K $和$ \ MATHCAL {S}^{ - 1} _K $作用于强对偶性数据$ D $。我们证明，如果$ d $是一个强（resp。\完整）双重性基准，则$ \ mathcal {s} _k（d）$和$ \ nathcal {s} _k^{ - 1}（d）（d）$也很强（resp。\ pounter）偶性数据。我们最终通过使用双重性函数$ f_d $在$ \ Mathcal {c} _ {\ Mathfrak {g}} $中介绍了仿生的cuspidal模块的概念，并开发了与Quiver affine Algebras相似的cuspidal模块理论，与Quiver hecke Hecke Hecke Hecke Algebra Algebra案例相似。
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $\mathcal{C}_{\mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $\mathcal{C}_{\mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Λ$ and $Λ^\infty$ introduced by the authors. We next define the reflections $\mathcal{S}_k$ and $\mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp.\ complete) duality datum, then $\mathcal{S}_k(D)$ and $\mathcal{S}_k^{-1}(D)$ are also strong (resp.\ complete ) duality data. We finally introduce the notion of affine cuspidal modules in $\mathcal{C}_{\mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.
