22 problems

The quantization of a classical superintegrable system can produce a quantum superintegrable system whose integrals of motion generate a quadratic associative algebra; the correspo…

Vershik's conjecture. For every $n\geqslant 3$, there is a finite dimensional quadratic algebra with $n$ generators given by

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 ${\cal E}_n^t$ 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…

Consider the algebra obtained from the Orlik–Solomon algebra by introducing two generators $x_{ij}$ and $y_{ij}$ for each pair $1\leq i\ne j\leq n$ and imposing the analogous relat…

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…

Let $\mathrm{QH}^*(Fl_n)$ be the quantum cohomology ring of the complete flag manifold $Fl_n$, and let the Dunkl elements be the pairwise commuting elements introduced by Fomin and…

Falk's converse conjecture. If $M$ is line-closed, then $A(M)$ is quadratic.

Let $G_\mathbb{F}(p)$ be the maximal pro-$p$ quotient of the absolute Galois group $G_\mathbb{F}$ of a field $\mathbb{F}$, and let the Jennings–Zassenhaus filtration of…

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 $n\in\mathbb{N}$, let $k\geqslant 2$, and let $\varphi_k$ denote the constant used in the paper's asymptotic bound. A $k$-step nilpotent algebra is a quadratic $\mathbb{K}$-alg…

A Koszul algebra is a graded algebra with the Koszul property. Let $A$ be a Koszul algebra, and let its Hilbert series be $H_A(t)$. Rational Hilbert series conjecture. The Hilbert…

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

Polyak's injectivity conjecture. The algebra homomorphism $\psi_n$ is injective.

Anick's conjecture. The Golod–Shafarevich estimate is attained in the case of quadratic algebras.

The generic superintegrable system on the 3-sphere has a quadratic algebra generated by the two-variable Wilson polynomials, and limiting processes are expected to produce the quad…

Anick's conjecture. The lower bound for the Hilbert series of $A$ is given by

Let the functions $\psi_{ij}$ be chosen as in Proposition 2.1(iii), with $K=\kappa\delta^2$ and $\lambda_{ij}=\delta\Lambda_{ij}$. Let $p_{ij}=p_{ji}=\kappa\Lambda_{ij}^{-2}$, and…