50 problems

Vogan's quantization conjecture. For each such cover $\widetilde{\mathcal{O}}$, there is a canonically defined completely prime primitive ideal $I_0(\widetilde{\mathcal{O}})$ in th…

For an associative algebra $\mathcal{A}$, let $\mathrm{Prim}^{\mathrm{fin}}(\mathcal{A})$ denote the set of primitive ideals of finite codimension. Let $\mathcal{U}_{0}$ and…

Let $V$ be an affine algebraic variety or algebraic group, and let ${\cal O}_{\bold q}(V)$ be a quantized coordinate ring of $V$. Let an appropriate group of automorphisms act on t…

Let $G$ be the algebraic group under consideration, let $H$ be a maximal torus acting on ${\cal O}_{q,p}(G)$ by winding automorphisms, and let…

Let $V$ be an affine algebraic variety, and let ${\cal O}_q(V)$ be a quantization of its coordinate ring. The quantized primitive ideal space quotient conjecture. The primitive ide…

Let $(A,G,\alpha)$ be a separable dynamical system, where $A$ is a $C^*$-algebra, $G$ is a locally compact group, and $\alpha$ is an action of $G$ on $A$. The system satisfies the…

Let ${\bf T}_n$ be the set of standard Young tableaux with $n$ boxes, let $I_T$ be the primitive ideal attached to $T$, let ${\mathcal V}_T$ be its orbital variety, and let…

For ${\mathfrak g}={\mathfrak s}{\mathfrak l}_n$, let ${\bf T}_n$ be the set of standard Young tableaux with $n$ boxes. To each $T\in{\bf T}_n$ attach a primitive ideal $I_T$ and a…

Let ${\bf T}_n$ be the set of standard Young tableaux with $n$ boxes, and consider the combinatorics of inclusions among primitive ideals and among closures of orbital varieties at…

Let ${\mathfrak g}={\mathfrak s}{\mathfrak l}_n$, and let the algebraic and geometric orders be the orders on standard Young tableaux induced respectively by inclusion of primitive…

Canonical left $\tilde{S}$-region conjecture. The ideal $J(\lambda)$ is completely prime.

Canonical cell conjecture. The element of minimal length is unique, and $J(\lambda)$ is a completely prime maximal ideal of ${\mathbf U}({^L \mathfrak{g}})$ whose associated variet…

Let $\W{-1}$ denote the negative Witt algebra, let $W$ denote the Witt algebra, let $\Vir$ denote the Virasoro algebra, and let $A_k$ and $\widetilde A_k$ denote the Weyl-type targ…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\Ua(\g)$ be its enveloping algebra. ACC conjecture. The algebra $\Ua(\g)$ satisfies the ascending chain condition on primitive…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\Ua(\g)$ be its enveloping algebra. Complete-primality conjecture. Every primitive ideal of $\Ua(\g)$ is completely prime. The…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$. Let $\operatorname{P.Prim}\operatorname{S}(\g)$ denote the Poisson primitive spectrum of the symmetric algebra and let…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\chi$ and $\eta$ be local functions on $\g$. Let $P(\chi)$ and $P(\eta)$ denote the associated Poisson primitive ideals, and l…

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\chi$ and $\eta$ be one-point local functions on $\g$. Let $P(\chi)$ and $P(\eta)$ denote their associated Poisson primitive i…

Let ${\mathfrak g}_{\mathbb C}$ be a reductive Lie algebra over ${\mathbb C}$ with Chevalley basis ${\mathcal B}$, let $R={\mathbb Z}[1/q\mid q\text{ is a bad prime for }\Phi]$, an…

Let $U(L_{\alpha,_i})$ be the enveloping algebra of the Takiff algebra $L_{\alpha,_i}$. Let $\mathfrak{I}(\mu)$ be the primitive ideals described in Theorem 4.3 for…

Let $G$ be a classical algebraic group with Lie algebra ${\mathfrak g}$, and let $U({\mathfrak g})$ be its enveloping algebra. Losev's orbit method gives a natural embedding from t…

Let $K$ be a field, let $\mathcal{G}$ be an ample groupoid, and let $u\in\mathcal{G}^{(0)}$. Write $A_K(\mathcal{G})$ for the Steinberg algebra, $\mathcal{G}_u$ for the isotropy gr…

Let $\mathfrak g$ be one of $V\!ir$, $W$, or $W_{\geq -1}$. For a local function $\chi\in\mathfrak g^*$, choose a polarization and form the annihilator of the induced module; write…

Let $W$ be the Witt algebra and let $\operatorname{U}(W)$ be its universal enveloping algebra. The Witt prime-to-primitive containment conjecture. Every proper prime ideal of…

Let $W$ be the Witt algebra and $W_{\geq -1}$ its subalgebra spanned by the basis elements of index at least $-1$. Let $\operatorname{U}(W)$ and $\operatorname{U}(W_{\geq -1})$ be…