8 problems

Let $K$ be an algebraically closed field of characteristic zero, let $\alpha_1,\alpha_2\in K$, and let $\mathcal{D}_4(\alpha_1,\alpha_2)$ be the quadratic $K$-algebra generated by…

Let $K$ be an algebraically closed field of characteristic zero, let $n\geq 3$, and let $\mathcal{E}_n$ be the quadratic $K$-algebra generated by elements $x_{ij}=-x_{ji}$ for…

Depth-one conjecture. The algebras $\mathcal E_n^!$ have depth $1$ for all $n\geq 2$.

Let $G$ be a non-exceptional indecomposable Coxeter group, and let $\mathcal{E}_G$ be its Fomin–Kirillov algebra. Finite-dimensionality conjecture. Up to isomorphism, the only fini…

Let $G=G(m,p,n)$ be a complex reflection group in the non-exceptional series, and let $\mathcal{B}_G$ be its elementary Fomin–Kirillov Nichols algebra. Finite-dimensionality conjec…

For each integer $n\geq 2$, let $\mathcal{E}_n$ denote the quadratic Fomin–Kirillov algebra associated with the symmetric group $S_n$. Majid's conjecture. The algebra…

Infinite-dimensionality conjecture. For every integer $n\geq 6$, one has

Let $n$ be a positive integer, let $\alpha_{ijk},\beta_{ijk},\gamma_{ij},\lambda_{ijkl}\in\{1,-1\}$, and let $A(\alpha,\beta,\gamma,\lambda)$ be the algebra generated by $x_{ij}$ f…