Characterize which quasi-cocommutative CTBs admit double structures

Determine whether an arbitrary quasi-cocommutative combinatorial twisted bialgebra can be equipped with a second coproduct making it a combinatorial twisted double bialgebra, and characterize such second coproducts when they exist.

Background

The paper introduces quasi-cocommutativity as a necessary condition for a combinatorial twisted bialgebra to admit a compatible second coproduct. It proves that several CTBs, including Exch and Exch⁺, are not quasi-cocommutative and therefore cannot be made into double bialgebras. For quasi-cocommutative CTBs, the paper constructs a coaction-like map and derives consequences, but it does not settle in general whether a compatible second coproduct exists. It does establish maximal sub-CTBs that can be made double in particular cases, such as Exch⁺, and proves uniqueness there.