2019年6月7日

# Relaxed highest-weight modules II: classifications for affine vertex algebras

• Kazuya Kawasetsu
• ,
• David Ridout

This is the second of a series of articles devoted to the study of relaxed<br />
highest-weight modules over affine vertex algebras and W-algebras. The first<br />
studied the simple &quot;rank-$1$&quot; affine vertex superalgebras<br />
$L_k(\mathfrak{sl}_2)$ and $L_k(\mathfrak{osp}(1\vert2))$, with the main<br />
results including the first complete proofs of certain conjectured character<br />
formulae (as well as some entirely new ones). Here, we turn to the question of<br />
classifying relaxed highest-weight modules for simple affine vertex algebras of<br />
arbitrary rank. The key point is that this can be reduced to the classification<br />
of highest-weight modules by generalising Olivier Mathieu&#039;s theory of coherent<br />
families. We formulate this algorithmically and illustrate its practical<br />
implementation with several detailed examples. We also show how to use coherent<br />
family technology to establish the non-semisimplicity of category $\mathscr{O}$<br />
in one of these examples.

リンク情報
arXiv
http://arxiv.org/abs/arXiv:1906.02935
URL
http://arxiv.org/abs/1906.02935v2