Polynomial fusion rings of W-extended logarithmic minimal models

Understand

The countably infinite number of Virasoro representations of the logarithmic minimal model LM(p,p') can be reorganized into a finite number of W-representations with respect to the extended Virasoro algebra symmetry W.

• Using a lattice implementation of fusion, we recently determined the fusion algebra of these representations and found that it closes, albeit without an identity for p>1.
• Here, we provide a fusion-matrix realization of this fusion algebra and identify a fusion ring isomorphic to it.
• We also consider various extensions of it and quotients thereof, and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings.