5 problems

Let the maps in Proposition and Theorem be the natural maps associated with the $\mathbb Z_3$-symmetric down-up algebra $\mathbb A=\mathbb A(\alpha,\beta,\gamma)$. Injectivity conj…

Extension conjecture. Theorem ${\rm{}}$ also holds when $\beta=-1$ and $\alpha\neq 2$. This would extend the noncommutative Auslander theorem to the remaining parameter cases for n…

Rigidity conjecture. The graded noetherian down-up algebras are rigid with respect to semisimple Hopf algebra actions.

The generalized grading conjecture. The corresponding algebra $S$ has a direct-sum algebra grading

The grading conjecture. The degree-zero component has basis