33 problems

Let $R_{\text{GGS}}$ be the matrix defined by … where the polynomials $P_{i,j,k,l}$ are uniquely determined by … Here $q=e^{\hbar}$, and the quantum Yang–Baxter equation is … while…

Gauge-element conjecture. In the case of $sl(n)$, there exists an element $g(x)\in \mathfrak{U}_q((sl(n))$ such that

Let $G$ and $G'$ be connected, simply connected semi-simple algebraic groups, and let $\psi:G'\times G'\to G$ be an admissible embedding. Let $\tau$ be the triple obtained from the…

Spectral-parameter 6-j-symbol conjecture. After introducing the spectral parameter, the resulting objects should be related to the 6-j symbols for quantum affine Kac–Moody algebras…

Let $R^{\mathrm{trig}}$ be the trigonometric quantum $R$-matrix, and let $R^{\mathrm{rat}}$ denote its $q\to1$ limit. A quantum integrable model has a $2\times2$ quantum Lax matrix…

Universal-algebra lift conjecture. Whenever an analogous relation occurs at the level of evaluated $R$-matrices, a similar relation exists at the level of universal algebras.

Let $\mathfrak{g}=sl(n+1)$ with $n\geq 1$, let $R(x)$ be the standard universal solution of the Quantum Dynamical Yang--Baxter Equation, and let $M(x)\in U_q(sl(n+1))$. For element…

For each $n$, let $QA_n$ be the quadratic algebra generated by the cohomology classes associated with the generators of the quasitriangular group, and let…

For each $n$, let ${\mathfrak{qtr}}_n$ be the Lie algebra generated by the elements $r_{ij}$, subject to the classical Yang–Baxter relations, and let ${\mathbf{QTr}}_n$ be the grou…

Let ${\mathbf{Tr}}_n$ and ${\mathbf{QTr}}_n$ be the triangular and quasitriangular groups, and let ${\mathfrak{tr}}_n$ and ${\mathfrak{qtr}}_n$ be their corresponding Lie algebras.…

Let $n$ and $k$ be coprime positive integers with $1\le k<n$, let $d\in\mathbb N$, and let $R_{n,k}(\mathcal E,\eta)$ be the corresponding elliptic solution of the Yang–Baxter equa…

Let a multi-colour braid-monoid algebra be a braid-monoid algebra whose strands or strings are assigned colours, and let $m$ denote the number of colours. Yang–Baxterization conjec…

A multi-colour braid-monoid algebra is the algebra generated by braid and monoid operators carrying one of several colours; Yang-Baxterization means constructing a spectral-paramet…

Level-two homology formula conjecture.

Free-rank conjecture. The rank of the free part of $H_n(R_{(m)})$ is $2^{m-1}$ for $n\geq m-1$, while

Third-homology formula conjecture.

Let $[? Let$[?

Let $N\in\mathbb{N}$, let $(1,R)\in YB^1(\mathsf{Match}^N)$, and let $X$ be a diagonal matrix in $\mathsf{Mat}^N(2,2)$. Define … For each $n$, let $\rho_n^R$ and $\rho_n^S$ denote…

Let $YB^1(\mathsf{Mat}^N)$ be the category of Yang–Baxter representations associated with $N\times N$ matrices, and let $YB^1(\mathsf{Matcha}^N)$ denote the subcategory associated…

The even part of the Haagerup subfactor planar algebra $H\mathcal{P}$ is a subfactor planar algebra lying in a one-parameter family of subfactor planar algebras. Haagerup family co…

Inequivalence conjecture. The $R$-matrices for $Osp_q(N|2m)$ and $Osp'_q(2m|2n)$, together with the corresponding quantum groups, are inequivalent.

Completeness conjecture. The elements $Q_k$ are complete: for every braid $X$ with $k$ strands, $Q_k(X)$ is a polynomial in the commuting variables $Q_1,\ldots,Q_k$ and the deforma…

Let $\lambda$ be an arbitrary weight and let $\Gamma$ be a reduced chain of roots corresponding to $\lambda$. Write $\mathcal{A}(\Gamma)$ for the associated quantum alcove model, a…

Let the $osp$-invariant Shankar–Witten (SW) type $\hat{\cal R}$-operator be expressed as a sum over $osp$-invariants $I_k$, and let the $osp$-invariant Faddeev–Tarasov–Takhtajan (F…

Superuniversality conjecture. All such models exhibit superdiffusion of the KPZ type in equilibrium states with unbroken symmetry.