论文标题
用于研究保形嵌入和折叠水平的新方法 - 代数
New approaches for studying conformal embeddings and collapsing levels for $W$--algebras
论文作者
论文摘要
在本文中，我们证明了一个普遍的结果，说在某些假设下，仿射顶点代数嵌入仿射$ w $中 - 当且仅当其中央指控重合时，代数才是保构。该结果扩展了我们先前获得的结果，在最小的仿射$ W $ - 代数。我们还发现有足够的条件，表明某些共形水平正在崩溃。这种新的条件使我们能够找到一些级别$ k $，其中$ w_k（sl（n），x，f）$ $ f $是挂钩或矩形类型时倒下的仿射部分。我们的方法可以应用于不可能的水平。特别是，我们证明了Creutzig在其offine vertex subalgebra的挂钩类型$ W $ -Algebra $ w_k（sl（n+m），x，f_ {m，n}）$中的共形嵌入中的猜想。令人惊讶的是，表明某些保形水平并没有崩溃的问题非常困难。在$ k $是可接受和保形的情况下，我们证明$ w_k（sl（n+m），x，f_ {m，n}）$不会崩溃。然后，通过从以前的论文中概括结构性嵌入的半简化结果，我们发现许多情况下，$ w_k（sl（n+m），x，x，f_ {m，n}）$是半简单的，作为其仿射子代理的模块，用于其正形级别，我们提供了explacicit decompocits。
In this paper we prove a general result saying that under certain hypothesis an embedding of an affine vertex algebra into an affine $W$--algebra is conformal if and only if their central charges coincide. This result extends our previous result obtained in the case of minimal affine $W$-algebras. We also find a sufficient condition showing that certain conformal levels are collapsing. This new condition enables us to find some levels $k$ where $W_k(sl(N), x, f )$ collapses to its affine part when $f$ is of hook or rectangular type. Our methods can be applied to non-admissible levels. In particular, we prove Creutzig's conjecture on the conformal embedding in the hook type $W$-algebra $W_k(sl(n+m), x, f_{m,n})$ of its affine vertex subalgebra. Quite surprisingly, the problem of showing that certain conformal levels are not collapsing turns out to be very difficult. In the cases when $k$ is admissible and conformal, we prove that $W_k(sl(n+m), x, f_{m,n})$ is not collapsing. Then, by generalizing the results on semi-simplicity of conformal embeddings from our previous papers, we find many cases in which $W_k(sl(n+m), x, f_{m,n})$ is semi-simple as a module for its affine subalgebra at conformal level and we provide explicit decompositions.
