论文标题
分类为可允许级别的Bershadsky-Polyakov代数的轻松最大重量模块
Classifying relaxed highest-weight modules for admissible-level Bershadsky-Polyakov algebras
论文作者
论文摘要
Bershadsky-Polyakov代数是与$ \ Mathfrak {SL} _3 $相关的仿射顶点代数的最低量子量化量子，而其简单的商在共同字段理论和弦理论中具有悠久的应用历史。因此，他们的表示理论非常有趣。在这里，我们针对所有可接受但不整合的水平进行了有限维度的重量空间的简单放松最高重量模块，并显着推广已知的最高重量分类[ARXIV：1005.0185，ARXIV：ARXIV：1910.13781]。特别是，我们证明了简单的Bershadsky-Polyakov代数，可容纳的非整合$ \ Mathsf {K} $在类别中始终是合理的$ \ Mathscr {O} $，而除非他们总是接纳nonsemisimples nonsemisimples nonsemisimples nonsemimples nosited nonsemimple saline dempled prainist bleas demble-weight deight deight deight deight dement weight dember-weight demand-weight除非$ \ nathsf {k’ \ Mathbb {z} _ {\ ge0} $。
The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $\mathfrak{sl}_3$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [arxiv:1005.0185, arxiv:1910.13781]. In particular, we prove that the simple Bershadsky-Polyakov algebras with admissible nonintegral $\mathsf{k}$ are always rational in category $\mathscr{O}$, whilst they always admit nonsemisimple relaxed highest-weight modules unless $\mathsf{k}+\frac{3}{2} \in \mathbb{Z}_{\ge0}$.
