论文标题
Kazhdan-Lusztig类别的张量结构$ \ Mathfrak {gl}（1 | 1）$
Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
论文作者
论文摘要
We show that the Kazhdan-Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure, and that its full subcategory $ \ MATHCAL {O} _K^{fin} $具有半密码Cartan subalgebra操作的对象是一个张量子类别。我们表明，每一个简单的$ \ wideHat {\ Mathfrak {gl}（1 | 1）} $ - $ kl_k $中的模块都具有$ \ Mathcal {o} _K^{fin} $中的投影封面，我们确定了所有涉及简单和投射的对象，涉及简单和投射的对象。然后，使用Knizhnik-Zamolodchikov方程，我们证明$ kl_k $和$ \ Mathcal {o} _K^{fin} $是刚性的。作为在$ \ mathcal {o} _k^{fin} $上的张量超级类别结构的应用，我们研究了仿射的某些模块类别lie superalgebra $ \ wideHat {\ mathfrak {s sl} {sl}（2 | 1）} $在$ 1 $ 1 $和$ - \ freac $ 1 $和$ - \ freac $ 1 $＆\ frac {1} $} $} $} $} $ {2} $} $ {2} = 2}。特别是，我们获得了$ \ wideHat {\ mathfrak {sl}（2 | 1）} $ - 级别$ - \ frac {1} {2} $的张量类别，其中包含放松的最高重量模块，并在频谱流下。
We show that the Kazhdan-Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure, and that its full subcategory $\mathcal{O}_k^{fin}$ of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple $\widehat{\mathfrak{gl}(1|1)}$-module in $KL_k$ has a projective cover in $\mathcal{O}_k^{fin}$, and we determine all fusion rules involving simple and projective objects in $\mathcal{O}_k^{fin}$. Then using Knizhnik-Zamolodchikov equations, we prove that $KL_k$ and $\mathcal{O}_k^{fin}$ are rigid. As an application of the tensor supercategory structure on $\mathcal{O}_k^{fin}$, we study certain module categories for the affine Lie superalgebra $\widehat{\mathfrak{sl}(2|1)}$ at levels $1$ and $-\frac{1}{2}$. In particular, we obtain a tensor category of $\widehat{\mathfrak{sl}(2|1)}$-modules at level $-\frac{1}{2}$ that includes relaxed highest-weight modules and their images under spectral flow.
