5 problems

Quantum-group interpolation conjecture. These standard combinatorial Hopf algebras should be incorporable into one-parameter families that contain an enveloping algebra and its dua…

Let $({\mathcal{H}},\Sigma)$ be a combinatorial Hopf algebra, and consider its Lie group of controlled characters. Regularity conjecture. With better estimates, or a refinement of…

Freeness conjecture. The algebra ${\bf PBT}^\sharp$ is free over the set ${\bf P}_T^\sharp$, where $T$ runs over trees with at least two nodes and such that the right subtree of th…

Triangularity conjecture. The matrix of ${{\bf S}^\sigma}^\sharp$ in the ${\bf S}$ basis is triangular. Its diagonal values are $1$ for elements of $X_n$ and $0$ for elements of…

Conjectural operad-algebra property. If such an operad ${\mathcal{X}}$ exists, then every cofree-cocommutative right-sided combinatorial Hopf algebra is an ${\mathcal{X}}$-algebra.