288 problems

Let $n$ be a positive integer, and let the $n$-th Weyl algebra be defined over a field of characteristic zero. Dixmier's conjecture. Every endomorphism of the $n$-th Weyl algebra i…

Gelfand–Kirillov conjecture. The skew field of fractions of $U$ should be a suitable Weyl field.

Noetherianity conjecture. The universal enveloping algebra

Let $A$ be a finitely presented graded algebra, and let $H_A(z)$ denote its Hilbert series. Govorov's conjecture. The Hilbert series $H_A(z)$ is rational. This conjecture is a hist…

Quantum Gel'fand–Kirillov conjecture. The skew field $\operatorname{Fract} A$ is isomorphic to the skew field of fractions of a quantum affine space over $K$.

Let $\Bbbk$ be an algebraically closed field of characteristic zero and let $q\in\Bbbk^\times$ be a nonroot of unity. For $n\geq 2$, let $\mathcal O_q(M_n(\Bbbk))$ denote the singl…

Wang–Zhang–Zhuang's classification conjecture. The families (I)–(VI) constitute all affine Hopf $k$-algebra domains of Gelfand–Kirillov dimension $2$, at least in the pointed case.

Let $A_{n,\mathbb{C}}$ be the Weyl algebra and let $P_{n,\mathbb{C}}$ be the commutative polynomial algebra equipped with its Poisson bracket. For an infinite prime $[p]$, let … be…

Kontsevich's conjecture. Every component of every such iterate is a noncommutative Laurent polynomial in $x$ and $y$.

Let ${\mathcal O}_q(M_{m,n})$ denote an algebra of quantum matrices, and let a torus act on it so that one can consider its torus-invariant prime ideals. A generating set is polyno…

Generalized Dixmier conjecture. Every endomorphism of $A_n(K)$ is an automorphism. This extends the first-Weyl-algebra problem to all Weyl algebras; the supplied text gives no reso…

Let $T_m(K)$ denote the algebra of upper triangular $m\times m$ matrices over a field $K$, and let $p$ be a multilinear polynomial in noncommutative variables. Fagundes–de Mello co…

Let $p$ be a multilinear polynomial in $m$ variables over a field $k$, and let $\Phi_p$ be the associated polynomial map on $M_n(k)$. Write $\mathfrak{sl}_n(k)$ for the set of trac…

Let $f(X,Y)\in\mathbb{C}[X,Y]$ be irreducible, and let $A_1=\mathbb{C}[x][\partial]$ be the first Weyl algebra. Consider pairs $(P,Q)\in A_1^2$ satisfying $f(P,Q)=0$, with the acti…

A quantized coordinate ring is a noncommutative algebra serving as a quantum analogue of the coordinate ring of an affine algebraic variety, and its prime spectrum consists of its…

Let $[a]\times[b]$ be the product of two chains, and let $\operatorname{NAR}$ denote noncommutative birational rowmotion on this poset, with labels taken in a noncommutative skew f…

Let $V$ be the defining space of the quantum matrix group, with coordinate functions $x^i_j$ and quantum diagonal subalgebra…

Cancellation conjecture.

Let $A$ be an Artin-Schelter regular algebra, and let $\nu_A$ denote its Nakayama automorphism. The homological determinant $\operatorname{hdet}(\nu_A)$ is the homological determin…

Let $\mathcal D_n={\mathbb C}[x_1,\ldots,x_n,\partial_1,\ldots,\partial_n]$ be the Weyl algebra, equipped with the filtration … where $a_\alpha\in{\mathbb C}[x_1,\ldots,x_n]$. Weak…

Kontsevich's conjecture. For all positive integers $r_1,r_2$ and all $m\geq 0$, the expressions

Let $k$ be the base field, let $q$ be the parameter, and let $\operatorname{\mathcal O}_q(M_{m,n}(k))$ denote the quantum coordinate ring of $m\times n$ matrices over $k$. A prime…

Let $K$ be a field and let $A$ be a finitely generated $K$-algebra that is a domain of finite GK dimension. Let $Q(A)$ be its quotient division algebra, and let $D$ be a division s…

Let $K$ be a field and let $A$ be a finitely generated $K$-algebra that is a domain of GK dimension $d$. A chain … of division subalgebras has stratiform length $m$ when each $D_i$…

Dubois-Violette's conjecture. Any $3$-regular multilinear form provides an algebra in $\mathcal{C}_2$.