17 problems

Let $E_{\tau,\eta}(\operatorname{SL}_{2})$ be the elliptic quantum group defined by the $RLL$ relations and determinant relation above, and let $L$ denote its $L$-operator. Let…

Let $L$ be a representation of the quantum $L$-operator algebra, and let $\hat L$ be a representation of the algebra defined by the corresponding $RLL$ relation. Let $\phi(z)$ and…

Let ${\bf g}$ be one of the twisted affine Lie algebra types $A_{2n}^{(2)}$ or $A_{2n-1}^{(2)}$, or let the type be $G_2^{(1)}$. Let ${\cal R}(\lambda)$ denote the universal dynami…

Let ${\bf g}$ be an affine Lie algebra, and consider the connection matrices for the solutions of the associated $q$-KZ equations in the vector representation. Jimbo–Miwa–Okado's f…

Representation-theoretic solvability conjecture. The representation theory of the elliptic quantum groups of both the vertex type ${\cal A}_{q,p}(\widehat{\operatorname{sl}}_N)$ an…

Let $\widehat{L}^+(v)\in\operatorname{End}(\mathbb{C}^N)\otimes U_{q,p}(\widehat{\mathfrak{sl}}_N)$ be the L-operator constructed from the half currents, and let $R^+(v,P)$ be the…

Elliptic quantum group formulation conjecture. The elliptic quantum groups $U_{q,p}(\widehat{\mathfrak g})$ and $U_{t_1,t_2,p}(\mathfrak{gl}_{tor})$ give alternative formulations o…

Elliptic quantum group and geometric representation conjectures. (1) Vertex operators of $U_{q,p}(\widehat{\mathfrak g})$ or $U_{t_1,t_2,p}(\mathfrak g_{tor})$ defined as intertwin…

Let ${\mathcal U}$ be the elliptic quantum toroidal algebra considered in the paper, and let ${\mathcal F}^{(0,-1)}_u$ be the level-(0,-1) representation on the vector space with b…

Elliptic Casimir monodromy conjecture. By analogy with the rational and trigonometric cases, the monodromy of the elliptic Casimir connection is described by the quantum Weyl group…

Let $U_{q,p}(\widehat{{\bf sl}}_N)$ be the dynamical elliptic quantum group used to construct elliptic weight functions of type $\widehat{{\bf sl}}_N$, and let elliptic stable enve…

Fix distinct $i,j\in\mathbf I$ and set $m=1-a_{ij}$. Define the elliptic Serre expression $\mathcal{S}(u,v)$ by the displayed formula in the paper, with…

Let $\mathcal{E}_{\hbar,\tau}(\mathfrak{g})$ be an elliptic quantum group and let $\operatorname{Rep}(\mathcal{E}_{\hbar,\tau}(\mathfrak{g}))$ be its category of representations. L…

Let $U_{q,p}([...??]$ be the elliptic quantum algebra with half currents assembled into the $L$-operator $[...??]$. For two auxiliary tensor components, write $[...??]$ and let…

Let ${\cal B}_{q,\lambda}(\widehat{sl}(M|N))$ be the elliptic quantum group associated with the affine Lie superalgebra $\widehat{sl}(M|N)$, and let $U_{q,p}(\widehat{sl}(M|N))$ be…

Let $U_{q,x}(\widehat{\mathfrak{sl}}_2)$ be the elliptic quantum group used to derive elliptic hypergeometric functions, and let ${}_{10}W_{9}$ denote Wilson's biorthogonal functio…

Let ${\bf g}$ be an untwisted affine Lie algebra or the twisted affine Lie algebra of type $A^{(2)}_2$, and let $U_{q,p}({\bf g})$ be the associated elliptic algebra with elliptic…