7 problems

Let $W$ be a Coxeter group and let $v$ be a vertex of its Bruhat graph. A Bruhat path from $v_0=\operatorname{id}$ to $v$ is called $\Psi$-generating when the Woronowicz symmetrisa…

Braided Brzezinski theorem. The variant of the Brzezinski theorem for braided Hopf algebras is valid.

Let $V$ be a vector space of dimension $N$, let $\rho$ be the $R$-matrix-induced structure on the braided exterior algebra $[...].$ For a long knot, let $J_{\rho}$ denote the assoc…

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 $q=\omega$ be a root of unity of order $N\geq 1$, let $H_{c,t}$ be the $4N$-dimensional Nichols f-algebra with scaling automorphism $\phi_t x_i=t_i x_i$, and let $R_{c,t}$ be t…

Let $V$ be a braided vector space of diagonal type, and let $\mathscr{B}(V)$ be its Nichols algebra. The root system of $\mathscr{B}(V)$ is the root system arising from its restric…

Let $V$ be a braided vector space of diagonal type with finite-dimensional Nichols algebra $\mathcal{B}(V)$. A pre-Nichols algebra of $V$ is a graded braided Hopf algebra of the fo…