Toroidal Grothendieck rings and cluster algebras

We study deformations of cluster algebras with several quantum parameters, called toroidal algebras, which naturally appear in the Grothendieck rings representations affine algebras. In this context, we construct and establish these are flat rings. prove that for a family monoidal categories $${\mathscr {C}}_1$$ simply-laced categorifying finite-type ring has natural structure algebra.