Irrational CFTs either fail to have only a finite number of modules of the chiral algebra, or the modules that arise are not semisimple.
We propose to control these difficulties by using a continuum limit from lattice models, in which the chiral algebra is replaced by a finite-dimensional associative algebra. The commutant of the latter is a “symmetry algebra”, and the use of this symmetry, and of categorical and functorial methods, is fruitful. In particular, from a class of models we obtain a ribbon Hopf algebra Morita equivalent to U_q(sl_2) for q certain roots of unity.
There is a natural notion of fusion for modules of the lattice version of the chiral algebra also, and the functorial relations with the symmetry algebra allow to obtain fusion rules for some indecomposable modules in the continuum limit.
Work done in collaboration with H. Saleur.