17 problems

Let $\mathbf{Vir}$ be the Virasoro algebra, and call a $\mathbf{Vir}$-module a Harish-Chandra module when it is a weight module with finite-dimensional weight spaces. Kac's conject…

Conjecture 1.2. There are no irreducible mixed modules over the Virasoro algebra.

Nonexistence conjecture. There are no simple mixed $\mathcal{V}$-modules.

Weaker density–cuspidality conjecture. If an irreducible module $V\in\mathfrak{k}({\mathcal B})$ has at least one finite-dimensional weight space, then $V$ is dense if and only if…

Classification conjecture. An irreducible module $V\in K_0({\mathcal B})$ is either a Loop module or is parabolically induced from a standard parabolic subalgebra of ${\mathcal B}$…

Fernando's conjecture. An irreducible module $V\in\mathfrak{k}({\mathcal B})$ is dense if and only if it is cuspidal.

Exponential tensor module conjecture. Every simple $d524$-module in $d51c^{\mathrm{irr}}$ is isomorphic to a simple subquotient of an exponential tensor module.

Let $\mathfrak{g}$ be an affine Kac–Moody algebra and let $V$ be an irreducible weight module. A module is non-dense when its support is a proper subset of a coset of the root latt…

Let $\mathfrak{g}$ be an affine Kac–Moody algebra, and let $V$ be a weight $\mathfrak{g}$-module. The module is non-dense when its support is a proper subset of a coset $\lambda+Q$…

Let $\mathfrak{g}$ be an affine Kac–Moody algebra with Cartan subalgebra $\mathfrak{h}$, root system $\Delta$, center $\mathbb{C}c$, and root lattice $Q$ generated by $\Delta$. A w…

Let $A_d=\mathbb{C}[t_1^{\pm1},\ldots,t_d^{\pm1}]$, let $\mathcal{W}_d=\operatorname{Der}(A_d)$ be the Witt algebra, and let a weight module mean a $\mathcal{W}_d$-module whose wei…

For a weight module, the Euler–Poincaré pairing is defined by the alternating sum of the corresponding extension groups, while the Dirac index pairing is defined using Dirac indice…

Higher-rank extension conjecture. The same construction should produce such families for an arbitrary $n\geqslant 2$.

Let $\mathfrak{g}$ be the Lie algebra under consideration, graded by $\mathbb{Z}^n$, and let $V$ be a mixed $\mathfrak{g}$-module: there are weights $\lambda$ and $\lambda+\alpha$…

Let $W_n$ be the Witt algebra and let $V$ be a mixed $W_n$-module, meaning that $V$ has both finite-dimensional and infinite-dimensional weight spaces in the same coset of its weig…

Nonexistence conjecture for irreducible mixed modules. There are no irreducible mixed modules over the Virasoro algebra.

Mazorchuk's conjecture. There are no irreducible mixed modules over the Virasoro algebra.