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.

中文翻译：

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仿射顶点代数在最小W-代数II中的共形嵌入：分解

我们提出了计算最小简单仿射W-代数\（{W_k（\ mathfrak {g}，\ theta）} \）作为其最大仿射子代数\（{\ mathscr {V} _k （\ mathfrak {g} ^ {\ natural}）} \）处于共形水平k，也就是说，每当Virasoro向量为\（{W_k（\ mathfrak {g}，\ theta）} \）和\（{ \ mathscr {V} _k（\ mathfrak {g} ^ \ natural）} \）一致。特别强调了仿射融合规则在确定分支规则中的应用。在几乎所有情况下，当\（{\ mathfrak {g} ^ {\ natural}} \）是半简单的Lie代数时，我们证明，对于合适的保形能级k，\（{W_k（\ mathfrak {G}，\ THETA）} \）是同构的延伸\（{\ mathscr {V} _K（\ mathfrak {G} ^ {\天然}）} \）由它的简单模块。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}）} \）。为了在保形的水平来分析更复杂的非简单的电流扩展，我们给出了简单的显式实现W¯¯代数\（{W_ {K}（\ mathit {SL}（4），\ THETA）} \）在ķ  = -8/3。正如[3]中的推测，我们证明\（{W_ {k}（\ mathit {sl}（4），\ theta）} \）与顶点代数\（{\ mathscr {R} ^ { （3）}} \），并使用筛选算子构造无限多个奇异矢量。我们还为顶点代数\（{V_k（\ mathit {sl}（n））} \\）和某些特定的\（{V_k（\ mathit {sl}（m \ vert n）），m \ ne n，m，n \ geq 1} \）在任意级别。