Selecta Mathematica

, Volume 24, Issue 3, pp 2455–2498 | Cite as

On the classification of non-equal rank affine conformal embeddings and applications

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


We complete the classification of conformal embeddings of a maximally reductive subalgebra \(\mathfrak {k}\) into a simple Lie algebra \(\mathfrak {g}\) at non-integrable non-critical levels k by dealing with the case when \(\mathfrak {k}\) has rank less than that of \(\mathfrak {g}\). We describe some remarkable instances of decomposition of the vertex algebra \(V_{k}(\mathfrak {g})\) as a module for the vertex subalgebra generated by \(\mathfrak {k}\). We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings \(A_1 \times A_1 \hookrightarrow C_3\) at level \(k=-1/2\), and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding \(A_1 \times D_4 \hookrightarrow C_8\) at level \(k=-1/2\) we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.


Conformal embedding Vertex operator algebra Non-equal rank subalgebra Howe dual pairs q-series identity 

Mathematics Subject Classification

Primary 17B69 Secondary 17B20 17B65 


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This work was done in part during the authors’ stay at Erwin Schrödinger Institute in Vienna (January 2017). D.A. and O. P. are partially supported by the Croatian Science Foundation under the Project 2634 and by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (KK. We thank the referee for his/her careful reading of the paper.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Dražen Adamović
    • 1
  • 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 RomaRomeItaly

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