Japanese Journal of Mathematics

, Volume 12, Issue 2, pp 261–315 | Cite as

Conformal embeddings of affine vertex algebras in minimal W-algebras II: decompositions

  • Dražen AdamovićEmail author
  • Victor G. Kac
  • Pierluigi Möseneder Frajria
  • Paolo Papi
  • Ozren Perše


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.

Keywords and phrases

conformal embedding vertex algebra W-algebra 

Mathematics Subject Classification (2010)

17B69 (primary) 17B20 17B65 (secondary) 


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Copyright information

© The Mathematical Society of Japan and Springer Japan KK 2017

Authors and Affiliations

  • Dražen Adamović
    • 1
    Email author
  • Victor G. Kac
    • 2
  • Pierluigi Möseneder Frajria
    • 3
  • Paolo Papi
    • 4
  • Ozren Perše
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia
  2. 2.Department of MathematicsMITCambridgeUSA
  3. 3.Politecnico di Milano, Polo regionale di ComoComoItaly
  4. 4.Dipartimento di MatematicaSapienza Università di RomaRomaItaly

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