128 problems

Let $f_1,\ldots,f_l,g\in\mathbb{F}\langle x_1,\ldots,x_d\rangle$ be noncommutative polynomials. The conjecture asserts that $g$ belongs to the two-sided ideal $(f_1,\ldots,f_l)$ if…

Let $W_{\geq -1}=\mathbb{C}[t]\partial$ and $W=\mathbb{C}[t,t^{-1}]\partial$ be the Witt Lie algebras. The conjecture is that the enveloping algebras…

For every finite-dimensional quantum complete intersection $A$ over an arbitrary field and every finite-dimensional left $A$-module $M$, if $\operatorname{Ext}^{1}_{A}(M,M)=0$ and…

Let $K$ be a field of characteristic zero. The Dixmier conjecture. Every endomorphism of the first Weyl algebra … is an automorphism. Here $A_1(K)$ is generated by $t,x$ with…

Linear independence conjecture. The subset $\{\langle f_i\rangle\}_{i=1}^r$ of $X(A)$ is $\mathbb Z$-linearly independent.

Let $\mathfrak{g}$ be the Lie algebra of a linear algebraic group over $\mathbb{C}$. For $r,s\in\mathbb{Z}_+$, let $\mathcal{D}_{r,s}=\mathcal{D}_r\otimes\mathbb{C}[y_1,\dots,y_s]$…

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…

A noetherian Hopf algebra is a Hopf algebra satisfying the ascending chain condition on left and right ideals. Brown-Goodearl conjecture. Every noetherian Hopf algebra is Artin–Sch…

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…

Kontsevich's conjecture. For any $r_1,r_2\in\mathbb{Z}_{>0}$, all these iterations are given by noncommutative Laurent polynomials in $x$ and $y$.

Non-toric NCCR extension conjecture. The statement of the theorem holds also if $\Lambda'$ is a non-toric NCCR.

Let $\mathbb{K}$ be a field, let $q\in\mathbb{K}^*$ be not a root of unity, and let ${\mathcal O}_q(M_n)$ be the quantisation of the ring of regular functions on $n\times n$ matric…

Goodearl's conjecture. If $R$ is an affine algebra of GK-dimension $2$, then its Jacobson radical $J(R)$ is nil.

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 $D_i$…

Let $R$ be an algebra over a field $K$, let $z$ be a free generator, and let $x_1,\dots,x_n$ be noncommuting free generators. Write $K\langle x_1,\dots,x_n\rangle$ for the free ass…

Let $A$ be a regular graded algebra, meaning that it has finite global dimension and satisfies the Gorenstein condition … The algebra $A$ is graded coherent if every finitely gener…

The $G_{j,\alpha}^{B_n}$ elementary-symmetric-function conjecture. For every $\alpha\in\mathbb{Q}$,

Let $H_n(u,t)$ be the generalized double affine Hecke algebra associated with an affine Dynkin diagram $D=\widetilde D_4,\widetilde E_6,\widetilde E_7$, or $\widetilde E_8$, with p…

Let $k$ be any field, and let $R\hookrightarrow S$ be an inclusion of $k$-algebras that splits as a map of $R$-modules. Simplicity descent conjecture. If $D_k(S)$ is a simple ring,…

Zeta-distribution main conjecture. There exist $f,g\in\Lambda$ such that $(\operatorname{rn}(f_ng_n^{-1}))_n$ is the zeta distribution $\mathcal{L}_S(G_\infty,M^\lor,1-k)$ and the…

Non-abelian Main Conjecture. The element $\delta_p(G_\infty,M,k)$ is induced by a generator

Let $C(U_{q,\Theta}(n))$ be the universal $C^*$-algebra generated by $\{\mathscr{D}^{-1}\}\cup\{V_{i,j}:1\leq i,j\leq n\}$ satisfying the defining relations above, and let…

Unobstructedness and smooth-centre conjecture. The point $[A]$ is smooth in $\mathcal{M}_d$, equivalently, $A$ is unobstructed; moreover, the dimension of the centre is constant in…

Inverse-image conjecture. For any word $w$ in $\mathcal{H}^0$ that cannot be written in the form $w_1\tau(w_2)$ for words $w_1,w_2$ in $\mathcal{H}^{2,3}$, we have