45 problems

Fusion products are constructed from tensor products of highest-weight cyclic modules for a current Lie algebra, using evaluation maps associated with a finite subset of…

Let $cmathfrak{g}$ be a semisimple complex Lie algebra, let $P^+$ be its dominant integral weights, and let $a_1,ldots,a_sincmathbb C$ be pairwise distinct with…

Let $\mathcal A_q$ be the current algebra and let $\mathcal O_q$ be the $q$-Onsager algebra. Baseilhac–Belliard's homomorphic-image conjecture. The algebra $\mathcal O_q$ is a homo…

Integral trace conjecture. We have

Let $F_{\lambda_1,\lambda_2}$ be the modules constructed in the paper from a pair of dominant integral weights $\lambda_1,\lambda_2$, and let $V(\lambda_i)_{c_i}$ denote the corres…

Let ${\mathfrak{g}}$ be a simple Lie algebra, let $\lambda=\lambda_1+\cdots+\lambda_n$ be a sum of dominant integral weights, and let $c_1,\ldots,c_n$ be pairwise distinct complex…

Let ${\mathfrak g}$ be a simple Lie algebra with fundamental representations $V^{(i)}$, $i=1,\dots,r$, and highest weight vectors $v^{(i)}\in V^{(i)}$. Let…

Let $\mathcal{A}^k_{k+s,k+r}$ be the vertex operator algebra generated by the currents $e(z)$ and $f(z)$, with $s$ and $r$ parameters and $k$ the representation level. A q-deformat…

Let $\mathfrak g$ be a Lie algebra, let $\pi_i$ be irreducible finite-dimensional $\mathfrak g$-modules, and let $Z=(z_1,\ldots,z_n)\in\Sigma\subset\mathbb C^n$ have pairwise disti…

Finite-resolution conjecture. For nonabelian ${\cal G}_0$, there should exist a similar finite resolution whose first differential is the direct sum of the screening charges.

Let $P^+$ be the set of dominant integral weights, and let $V(\gamma)$ denote the irreducible highest weight representation of highest weight $\gamma$. For pairs of dominant weight…

Let $L$ be the Lie algebra and $\mathcal{OB}$ the interpolating category described above. For each $n \in \mathbb{N}$, let $\operatorname{Curr}(I_n)$ be the functor from the catego…

Bracket-width conjecture. The bracket width of $\mathfrak g\otimes_{G_a}G_a[[t]]$ is equal to $2$ if $\mathfrak g$ is of type $\mathrm A_n$ or $\mathrm C_n$ $(n\ge 2)$, and to $1$…

Evaluation-point independence conjecture. As a $\mathfrak g[t]$-module, the fusion product $V(\lambda_1)^{a_1}\ast\cdots\ast V(\lambda_k)^{a_k}$ is independent of the points of eva…

Let $W_1,\dots,W_r$ be the modules considered in the preceding construction, let $\mathcal{A}(\lambda_1,\dots,\lambda_r)$ be the associated parameter algebra, and let…

Let $P^+$ be the dominant integral weights of $cmathfrak{sl}_n$, let $cmathcal{P}_{clambda}$ be the set of decompositions of $clambdain P^+$ into two dominant weights, and let…

Let $cmathfrak{g}$ be a semisimple complex Lie algebra, let $P^+$ denote its dominant integral weights, and fix $clambda_1,clambda_2in P^+$. Put $clambda=clambda_1+clambda_2$, and…

Let $cmathfrak{g}$ be a semisimple complex Lie algebra, let $cmathfrak{g} cotimes cmathbb C[t]$ be its current algebra, and let $V(clambda_1)_{a_1} cast V(clambda_2)_{a_2}$ denote…

Local-product and fusion-product conjecture. The natural surjection from the local product to the fusion product is an isomorphism:

Let $\mathcal A_q$ be the current algebra, let $\mathcal O_q$ be the q-Onsager algebra with generators $A,B$, let $\{\tilde g_{k+1}\}_{k\in\mathbb N}\subseteq\mathcal J$ be the seq…

Let $\mathcal O_q$ be the q-Onsager algebra with generators $A,B$, and let $\mathcal J$ be its two-sided ideal generated by $A$ and $B$. Lower-generator uniqueness conjecture. Ther…

Let $J$ be the two-sided ideal of $\mathcal A_q$ generated by $\mathcal W_0$ and $\mathcal W_1$, and let $Z(\mathcal A_q)$ denote the center. Ideal–center direct-sum conjecture. Th…

Let $\mathcal A_q$ be the current algebra, let $Z(\mathcal A_q)$ denote its center, let $\Delta_{k+1}\in Z(\mathcal A_q)$, and let $\langle\mathcal W_0,\mathcal W_1\rangle$ be the…

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 $\lambda\in P^+$, let $\boldsymbol\omega\in\mathcal P_q^+$ be preminimal of order $2$, and let $L(\boldsymbol\omega)$ be the graded limit of $V_q(\boldsymbol\omega)$. Let…