A vertex operator algebra structure of degenerate minimal models, I

We study fusion rings for degenerate minimal models (p = q) for N = 0, N = 1 and N = 2 (super)conformal algebras. In the first part we consider a family of modules for the Virasoro vertex operator algebra L(1, 0), and show that a fusion ring of the family is isomorphic to a Grothendieck ring Rep(sl(2,C)). In the second part, we used similar methods for the family of modules for N = 1 Neveu Schwarz vertex operator superalgebra L( 3 2 , 0) and obtain a fusion ring isomorphic to Rep(osp(1|2)). Finally, in the N = 2 case we shall study vertex operator superalgebra L(3, 0, 0).