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Fusion of irreducible modules in WLM(p,p')

Based on symmetry principles, we derive a fusion algebra generated from repeated fusions of the irreducible modules appearing in the W-extended logarithmic minimal model WLM(p,p'). In addition to the irreducible modules themselves, closure of the commutative and associative fusion algebra requires the participation of a variety of reducible yet indecomposable modules. We conjecture that this fusion algebra is the same as the one obtained by application of the Nahm-Gaberdiel-Kausch algorithm and find that it reproduces the known such results for WLM(1,p') and WLM(2,3). For p>1, this fusion algebra does not contain a unit. Requiring that the spectrum of modules is invariant under a natural notion of conjugation, however, introduces an additional (p-1)(p'-1) reducible yet indecomposable rank-1 modules, among which the identity is found, still yielding a well-defined fusion algebra. In this greater fusion algebra, the aforementioned symmetries are generated by fusions with the three irreducible modules of conformal weights Delta_{kp-1,1}, k=1,2,3. We also identify polynomial fusion rings associated with our fusion algebras.