Communications in Mathematical Physics

, Volume 172, Issue 3, pp 623–659 | Cite as

On the classification of modular fusion algebras

  • Wolfgang Eholzer


We introduce the notion of (nondegenerate) strongly-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group Γ=SL(2,ℤ) whose kernel contains a congruence subgroup. Furthermore, nondegenerate means that the conformal dimensions of possibly underlying rational conformal field theories do not differ by integers. Our main result is the classification of all strongly-modular fusion algebras of dimension two, three and four and the classification of all nondegenerate strongly-modular fusion algebras of dimension less than 24. We use the classification of the irreducible representations of the finite groups\(SL(2,\mathbb{Z}_{p^\lambda } )\), wherep is a prime and λ a positive integer. Finally, we give polynomial realizations and fusion graphs for all simple nondegenerate strongly-modular fusion algebras of dimension less than 24.


Neural Network Statistical Physic Field Theory Positive Integer Complex System 
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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Wolfgang Eholzer
    • 1
    • 2
  1. 1.Max-Planck-Institut für Mathematik BonnBonnGermany
  2. 2.Physikalisches Institut der Universität BonnBonnGermany

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