This is the second of a series of papers devoted to the study of relaxed highest-weight modules over affine vertex algebras and W-algebras. The first [K. Kawasetsu and D. Ridout, Relaxed highest-weight modules I: Rank $1$ cases, Commun. Math. Phys. 368 (2019) 627–663, arXiv:1803.01989 [math.RT]] studied the simple “rank-$1$” affine vertex superalgebras $L_k({𝔰𝔩}_2)$ and $L_k(𝔬𝔰𝔭(1|2))$, with the main results including the first complete proofs of certain conjectured character formulae (as well as some entirely new ones). Here, we turn to the question of classifying relaxed highest-weight modules for simple affine vertex algebras of arbitrary rank. The key point is that this can be reduced to the classification of highest-weight modules by generalizing Olivier Mathieu’s coherent families [O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier $($Grenoble$)$ 50 (2000) 537–592]. We formulate this algorithmically and illustrate its practical implementation with several detailed examples. We also show how to use coherent family technology to establish the non-semisimplicity of category $\mathcal{𝒪}$ in one of these examples.

这是专门研究仿射顶点代数和 W 代数上的松弛最高权重模块的系列论文中的第二篇。第一个 [K. Kawasetsu 和 D. Ridout，放松的最高权重模块 I：排名$1$案例，交流。数学。物理。 368 (2019) 627–663, arXiv:1803.01989 [math.RT]] 研究了简单的“rank-$1$” 仿射顶点超代数${\mathrm{大号}}_{ķ}({𝔰𝔩}_2)$和${\mathrm{大号}}_{ķ}(𝔬𝔰𝔭(1|2))$，主要结果包括某些猜想字符公式的第一个完整证明（以及一些全新的公式）。在这里，我们转向为任意等级的简单仿射顶点代数分类松弛的最高权重模块的问题。关键是通过推广 Olivier Mathieu 的连贯族 [O. Mathieu，不可约重量模块的分类，Ann。研究所。傅立叶 $($格勒诺布尔$)$ 50 (2000) 537–592]。我们在算法上制定了这个算法，并用几个详细的例子来说明它的实际实现。我们还展示了如何使用相干族技术来建立类别的非半简单性$\mathcal{𝒪}$在这些示例之一中。