104 problems

Kashiwara crystal conjecture. should have the structure of a Kashiwara crystal isomorphic to $B(\lambda)$, the crystal of the integrable highest weight module of the quantized enve…

Rigged configuration virtualization conjecture. For all $\mathfrak{g}$ of symmetrizable type, there exists a simply-laced type $\widehat{\mathfrak{g}}$ and diagram folding $\phi$ w…

Let $\mathfrak g$ be the Kac–Moody Lie algebra under consideration and let $d$ be the imaginary root in the example, with $d=(n,n,1)$. Write $\mathfrak g_d$ for the root space of d…

Kac–Moody extension conjecture. The analogues of the equations giving the Chevalley formulae for $\lambda$ and $-\lambda$ hold for $W$.

Let $\mathfrak g$ be a Kac–Moody algebra, let $\Lambda_1,\dots,\Lambda_n$ be dominant integral weights, let $z_1,\dots,z_n$ be points, and let $\boldsymbol b$ be symmetric paramete…

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody Lie algebra, let $W_{\mathfrak{g}}$ be its Weyl group, and let $w\in W_{\mathfrak{g}}$. Write ${\mathcal U}_q^\pm[w]$ for the quantu…

Affine Kac–Moody chiral-algebra conjecture. The corresponding chiral algebra is

Direct-summand conjecture. For $0<|m|\leq 2$, the maximal submodule $U(m)$ is $\{0\}$, $V(m)=V^{\Lambda_m}$ is a direct summand of ${\mathcal F}ib(m)$, and

Let $J$ be a finite subset of the real indices, and let ${\frak g}_{0}^{(J)}$ and ${\frak g}_{-}^{(J)}$ be the subalgebras in the triangular decomposition associated with $J$. For…

Suppose that $\tilde{\Phi}$ is not of type $\mathsf{B}\mathsf{C}$. Let $\tilde{\mathbf A}$ be the Cartan matrix whose realization is given by the vector space ${}^e\!{\mathfrak s}$…

Invariant-factor conjecture. Its invariant factors are

All-exponents conjecture. The elementary divisors of this form are

First Kac conjecture. For any $\alpha\in Q_+$,

Conjectural character formulas. The character formula known for the part

Chain-independence conjecture. The colored directed graph defined by the action of root operators on the admissible subsets corresponding to any $\lambda$-chain does not depend on…

Let \text{bf g} be a symmetrizable Kac-Moody algebra, let $G$ be the (minimal) Kac-Moody group, and let $M_{drr}$ be the monoid associated to the category of integrable…

Let \text{bf g} be a symmetrizable Kac-Moody algebra, and let the Tannaka monoid be associated to the category of integrable \text{bf g}-modules with point-separating integ…

Canonical–global basis conjecture. The canonical bases thus obtained coincide with the global bases constructed by Jeong, Kang, and Kashiwara.

Kac's conjectures. For every root $d\in\mathbf{N}I$, there exists a polynomial $P_d(t)$ with nonnegative integer coefficients such that

Let $\mathbb{X}_{\mathbf{p},\underline{\lambda}}$ be a weighted projective line, let $\mathfrak{Lg}$ be its corresponding loop Kac–Moody algebra, and use the identification…

Global sections equivalence conjecture. The functor

At the critical level $k=-\check{h}$, let $\chi\in\mathfrak{h}^{\sigma,*}$ be a highest weight for the twisted affine Lie algebra $\widehat{\mathfrak{g}}^{\sigma}$, and let…

Let $M_{0,\lambda}(d,v)$ be the quiver variety in question, and set … Let $(L_d)_0$ denote the $0$-weight space of the Kac–Moody algebra associated to the quiver, for the represent…

Minimal-odd-root Dynkin diagram conjecture. There always exists exactly one Dynkin diagram with the minimal number of odd roots satisfying the definition of a hyperbolic Kac–Moody…

Let ${\mathfrak{e}}_{10}$ be the hyperbolic Kac–Moody algebra, and let $\mathcal{U}_{2}^{{\mathbbm{Z}}_{n}}$ be the automorphism defining the orbifold action. Let…