The B(∞)B(∞) crystal for a family of generalized quantum groups

In [1] Bozec gave a definition of generalized quantum groups that extends the usual definition of quantum groups to finite quivers with loops at vertices, and in [3] he introduced a theory of generalized crystals   for this new family of Hopf algebras. We explicitly characterize the generalized crystal B(∞)B(∞) associated to a certain family of comet-shaped quivers with multiple loops by providing a complete set of relations among the Kashiwara operators themselves.