Existence of coproduct/Hopf algebra structures for quantum toroidal algebras

Establish whether quantum toroidal algebras U_{q,tor}(X_n^{(r)}) admit genuine algebraic coproducts or Hopf algebra structures (beyond topological coproducts), across all types X_n^{(r)}. Clarify the precise conditions and constructions under which such structures exist and characterize their compatibility with known subalgebras and gradings.

Background

The paper constructs a new topological coproduct for quantum toroidal algebras that yields a well-defined tensor product on the category of integrable representations, addressing the lack of monoidal structure. However, the existence of genuine algebraic coproducts or full Hopf algebra structures for quantum toroidal algebras remains unresolved in general.

Resolving this would place quantum toroidal algebras on par with finite and affine quantum groups, where Hopf structures underpin braidings, tensor categories, and many applications in topology and integrable systems.