6 problems

Assume that $q$ is not a root of unity, that $(q,t_1,t_2)$ satisfies $t_1t_2q^n=1$ for some integer $n\geq 1$, and that $(1-t_1)(1-t_2)\ne 0$. Let $Y_n$ be the $4n$-dimensional rig…

Let $K$ be a finite-dimensional cosemisimple Hopf algebra, let $V\in {}^{K}_{K}\mathcal{YD}$ be a Yetter–Drinfeld module such that $\dim \mathscr{B}(V)<\infty$, and let…

Let $N_t$ be an object in the class whose existence is asserted, let $V$ be the associated Yetter–Drinfeld module, and let $V_0$ denote its degree-zero component. For $k\ge 2$, let…

Let $n\geq 5$. Let $a\sigma\in \mathbb W_n$, where $\sigma\in\mathbb S_n$ is nontrivial and $a\in H_n$. For the conjugacy class $\mathcal{O}_{a\sigma}^{\mathbb W_n}$ and an associa…

Let $M\langle(xy,x)\rangle$, $M\langle(y,xy)\rangle$, and $W^{b_1,-1}$ denote the Yetter–Drinfeld modules introduced in the paper, and let $\mathfrak{B}(-)$ denote their Nichols al…

Equality conjecture. For any $r$ relatively prime to $\exp(S_n)$ and $V\in{}^{S_n}_{S_n}\mathcal{YD}$, one has