13 problems

Let ${\cal B}_n^t$ be the quotient of the braid algebra ${\cal B}_n$ with parameters $t_{ij}$, and let $\mathbb{Q}[t]=\mathbb{Q}[t_{ij},\ 1\leq i<j\leq n]$. Finite-generation conje…

Let ${\cal E}_n^0$ be the quadratic algebra, let $Z_{n,k}$ be the sets of normal monomials defined in the paper, and write $H(A;t)$ for the Hilbert polynomial of a graded object…

Let ${\cal E}_{n,k}^0$ denote the homogeneous degree-$k$ component of the quadratic algebra ${\cal E}_n^0$, and let $n$ and $k$ be positive integers. Dimension formula conjecture.…

Let ${\cal E}_n^t$ be the quadratic algebra with central parameters $t_{ij}=[ij]^2$, and let $\mathbb{Z}[t]=\mathbb{Z}[t_{ij},\ 1\leq i<j\leq n]$. Finite-generation conjecture. The…

Let $\widetilde{\cal G}_n^0$ be the quotient of $\widetilde{\cal G}_n$ by the relations $[ij]^2=0$. Let $\widetilde\theta_1,\ldots,\widetilde\theta_n$ be the Dunkl elements, and le…

For each $n$, let $QA_n$ be the quadratic algebra generated by the cohomology classes associated with the generators of the quasitriangular group, and let…

Nonnegativity conjecture. If $S_n/I$ is a quadratic algebra, then $e_i(I)\ge0$ for all $i$.

Quadraticity–quadratic Gröbner basis conjecture. The algebra $\operatorname{A}(X_{n,d}^G)$ is quadratic if and only if $\operatorname{I}(X_{n,d}^G)$ has a quadratic Gröbner basis.

The three-dimensional restriction conjecture. The algebra $\operatorname{A}(X_{n,d}^G)$ is quadratic if and only if, for all $i<j<k$, the algebra…

Let $\Delta$ be the discriminant algebra operation, sending rank-$n$ algebras $\mathcal{A}\to\mathcal{B}$ to quadratic algebras $\mathcal{A}\to\Delta_{\mathcal{B}/\mathcal{A}}$. Le…

Let $A=\bigoplus_i A_i$ be an Artinian Gorenstein algebra presented by quadrics over a field $\mathbb{K}$ of characteristic zero. The Weak Lefschetz Property means that there exist…

Let $A$ be an Artinian Gorenstein algebra of socle degree at least three, presented by quadrics over a field $\mathbb{K}$ of characteristic zero. For $L\in A_1$, write…

Let $n\in{\mathbb N}$ with $n\geqslant 3$. Consider a generic quadratic ${\mathbb C}$-algebra, meaning a quadratic algebra in $n$ generators whose defining relations are generic in…