We present methods for computing the explicit decomposition of the minimal simple affine W-algebra \({W_k(\mathfrak{g}, \theta)}\) as a module for its maximal affine subalgebra \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) at a conformal level k, that is, whenever the Virasoro vectors of \({W_k(\mathfrak{g}, \theta)}\) and \({\mathscr{V}_k(\mathfrak{g}^\natural)}\) coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when \({\mathfrak{g}^{\natural}}\) is a semisimple Lie algebra, we show that, for a suitable conformal level k, \({W_k(\mathfrak{g}, \theta)}\) is isomorphic to an extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) by its simple module. We are able to prove that in certain cases \({W_k(\mathfrak{g}, \theta)}\) is a simple current extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\). In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra \({W_{k}(\mathit{sl}(4), \theta)}\) at k = −8/3. We prove, as conjectured in [3], that \({W_{k}(\mathit{sl}(4), \theta)}\) is isomorphic to the vertex algebra \({\mathscr{R}^{(3)}}\), and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra \({V_k (\mathit{sl}(n))}\) at certain admissible levels and for \({V_k (\mathit{sl}(m \vert n)), m\ne n, m,n\geq 1}\) at arbitrary levels.

我们提出计算最小简单仿射W代数\({W_k(\mathfrak{g}, \theta)}\)的显式分解作为其最大仿射子代数\({\mathscr{V}_k (\mathfrak{g}^{\natural})}\)在共形水平k，即，每当\({W_k(\mathfrak{g}, \theta)}\)和\({ \mathscr{V}_k(\mathfrak{g}^\natural)}\)一致。特别强调仿射融合规则在分支规则确定中的应用。在几乎所有情况下，当\({\mathfrak{g}^{\natural}}\)是半单李代数时，我们表明，对于合适的共形水平k，\({W_k(\mathfrak{g}, \ theta)}\)通过其简单模块与\({\mathscr{V}_k(\mathfrak{g}^{\natural})}\)的扩展同构。我们能够证明在某些情况下\({W_k(\mathfrak{g}, \theta)}\)是\({\mathscr{V}_k(\mathfrak{g}^{\自然的}）}\）。为了分析共形水平上更复杂的非简单电流扩展，我们提出了k处简单W代数\({W_{k}(\mathit{sl}(4), \theta)}\)的显式实现 =−8/3。正如[3]中的猜想，我们证明\({W_{k}(\mathit{sl}(4), \theta)}\)同构于顶点代数\({\mathscr{R}^{ (3)}}\)，并使用筛选算子构造无限多个奇异向量。我们还在某些允许水平上为顶点代数\({V_k (\mathit{sl}(n))}\)和\({V_k (\mathit{sl}(m \) vert n)), m\ne n, m,n\geq 1}\)在任意级别。