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Principal subspaces for the affine Lie algebras in types D, E and F

Abstract

We consider the principal subspaces of certain level \(k\geqslant 1\) integrable highest weight modules and generalized Verma modules for the untwisted affine Lie algebras in types D, E and F. Generalizing the approach of G. Georgiev we construct their quasi-particle bases. We use the bases to derive presentations of the principal subspaces, calculate their character formulae and find some new combinatorial identities.

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Fig. 1
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Notes

  1. In contrast with [21] and [23, Table Fin], we reverse the labels in the Dynkin diagram of type \(C_l\) in Fig. 1, so that the root lengths in the sequence \(\alpha _1,\ldots ,\alpha _l\) decrease for all types of \(\mathfrak {g}\), thus getting a simpler formulation of Theorem 3.1.

  2. Note that the quasi-particles of color 1 in type \(E_7\) correspond, with respect to the aforementioned identification, to the quasi-particles of color 7 in type \(E_8\); see Fig. 1.

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Acknowledgements

The authors would like to thank Mirko Primc for useful discussions and support. Also, we would like to thank the anonymous referees for the valuable comments and suggestions which helped us to improve the manuscript. This work has been supported by Croatian Science Foundation under the project UIP-2019-04-8488. The first author is partially supported 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 (Grant KK.01.1.1.01.0004).

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Correspondence to Slaven Kožić.

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Butorac, M., Kožić, S. Principal subspaces for the affine Lie algebras in types D, E and F. J Algebr Comb 56, 1063–1096 (2022). https://doi.org/10.1007/s10801-022-01146-x

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Keywords

  • Principal subspaces
  • Combinatorial bases
  • Combinatorial identities
  • Quasi-particles
  • Vertex operator algebras
  • Affine Lie algebras

Mathematics Subject Classification

  • Primary 17B67
  • Secondary 05A19
  • 17B69