15 problems

Quantum truncated PBW basis conjecture. The monomials

Classical truncated PBW basis conjecture. $S(n,d)$ has a ${\mathbb Q}$-basis consisting of all ordered monomials in $G$ of total degree at most $d$.

WG-basis conjecture. The ordered monomials in the three blocks $W^-<G<W^+$ form a PBW basis; this is the $WG$-basis.

Let $O_q$ be the $q$-Onsager algebra and let $mathcal A_q$ be the alternating central extension associated with it. Alternating central-extension conjecture. On the basis of suppor…

Let $\qbA{}\simeq\mathcal{K}_t(\catC_{\mathfrak{g}}^0)$, let $\mathtt{b}\in\bg^+$, and let $\qbA{}(\mathtt{b})$ be the associated subalgebra. The monoidal categorification conjectu…

Let $\mathtt{b}\in\bg^+$, let $\pd=(\textbf{\textit{i}},\xi)$ be a PBW datum of $\mathtt{b}$, and let $\qbA{}(\mathtt{b})$ be the associated subalgebra of the bosonic extension. Th…

Let $O_q$ be the $q$-Onsager algebra, with alternating generators $\{W_{-i}\}_{i\in\mathbb N}$, $\{\tilde G_{j+1}\}_{j\in\mathbb N}$, and $\{W_{k+1}\}_{k\in\mathbb N}$. A linear or…

Let $\mathcal A_q$ be the current algebra over $\mathbb F$, with generators $\mathcal W_{-k}$, $\mathcal W_{k+1}$, $\mathcal G_{k+1}$, and $\mathcal{\tilde G}_{k+1}$ for…

Let $cmathcal{A}(N)$ be the algebra generated by the modes $X^{(i)}_n$ acting on the Fock module $cmathcal{F}_{\vec{u}}$, and let $Ket{X_{\vec{\lambda}}}$ and…

PBW basis conjecture. These vectors induce a Poincare–Birkhoff–Witt-type basis of $\tilde{{\cal A}}_q^{\{\delta\}}$, called the $WG$-basis.

Let $\mathfrak{g}=\mathfrak{b}^-\oplus\mathfrak{n}^+$ be a symmetrizable Kac-Moody Lie algebra with Weyl group $W$. For $w,w'\in W$, define the bi-Schubert algebra … Here $U_q(w)$…

Affine PBW-basis conjecture. PBW-type bases can be constructed in affine type using methods similar to those in the cited work.

PBW-basis coincidence conjecture. These PBW bases should coincide.

Let $\mathfrak{g}$ be a Lie algebra, and let $\rho_{\mathbf{i}}$, $\pi_{\mathbf{i}}$, $\xi_i$, and the matrix entries $\rho_{\mathbf{i}}(e_i)_{AB}$ and…

Let $V(\lambda)$ be an irreducible highest-weight module, let $W$ be the Weyl group, and let $V_w(\lambda)\hookrightarrow V(\lambda)$ be a Demazure module for $w\in W$. For special…