论文标题
张量类别$ kl_k（\ mathfrak {sl} _ {2n}）$通过最小仿射$ w $ -algebras在不可加入的级别$ k = - \ frac {2n+1} {2n+1} {2} {2} $
Tensor category $KL_k(\mathfrak{sl}_{2n})$ via minimal affine $W$-algebras at the non-admissible level $k =-\frac{2n+1}{2}$
论文作者
论文摘要
我们证明，$ kl_k（\ Mathfrak {sl} _m）$是一个半简单，刚性编织的张量类别，甚至所有$ m \ ge 4 $，$ k = - \ \ \ \ \ frac {m+1} {2} {2} $从arxiv：2102985 $ $ m = 4 $ $ m = 4 $。此外，$ kl_k（\ mathfrak {sl} _m）$中的所有模块都是简单的，都是简单的，它们出现在共形嵌入$ \ mathfrak $ \ mathfrak {gl} _m \ hookrightArrow \ hookrightArrow \ mathfrak \ mathfrak {sl} _ {m+1} $ at c = - {M+k = - ARXIV：1509.06512。为此，我们诱导地确定最小仿射$ W $ -Algebra $ w_ {k-1}（\ Mathfrak {sl} _ {m+2}，θ）$是$ l_ {k}（k k}（\ mathfrak {slfrak {sl} {sl} _m _m）的简单电流扩展排名一个Heisenberg顶点代数，$ \ Mathcal m $ singlet顶点代数为$ c = -2 $。该证明使用了先前从ARXIV的单元代数的张量类别获得的结果：2202.05496。我们还为$ w_ {k-1}（\ mathfrak {sl} _ {m+2}，θ）$的所有不可约的普通模块分类。 $ w_ {k-1}类别的半简单部分（\ Mathfrak {sl} _ {m+2}，θ）$ - 模块来自$ kl_ {k-1}（\ mathfrak {slfrak {sl} _ {m+2}）$，但使用量子hamilton nodem norder nord $ node coldober and $ node coldober contrair and colder contrae $ w。
We prove that $KL_k(\mathfrak{sl}_m)$ is a semi-simple, rigid braided tensor category for all even $m\ge 4$, and $k= -\frac{m+1}{2}$ which generalizes result from arXiv:2103.02985 obtained for $m=4$. Moreover, all modules in $KL_k(\mathfrak{sl}_m)$ are simple-currents and they appear in the decomposition of conformal embeddings $\mathfrak{gl}_m \hookrightarrow \mathfrak{sl}_{m+1} $ at level $ k= - \frac{m+1}{2}$ from arXiv:1509.06512. For this we inductively identify minimal affine $W$-algebra $ W_{k-1} (\mathfrak{sl}_{m+2}, θ)$ as simple current extension of $L_{k}(\mathfrak{sl}_m) \otimes \mathcal H \otimes \mathcal M$, where $\mathcal H$ is the rank one Heisenberg vertex algebra, and $\mathcal M$ the singlet vertex algebra for $c=-2$. The proof uses previously obtained results for the tensor categories of singlet algebra from arXiv:2202.05496. We also classify all irreducible ordinary modules for $ W_{k-1} (\mathfrak{sl}_{m+2}, θ)$. The semi-simple part of the category of $ W_{k-1} (\mathfrak{sl}_{m+2}, θ)$-modules comes from $KL_{k-1}(\mathfrak{sl}_{m+2})$, using quantum Hamiltonian reduction, but this $W$-algebra also contains indecomposable ordinary modules.
