72 problems

Let $B_4$ be the braid group on four strands, and let $\mathrm{X}^{(2)}_{\gamma,\delta}$ be the set on which the induced braid-group action is defined. Assume Assumption. Transitiv…

Let $\mathfrak h$ be the Lie algebra occurring in the classical dynamical Yang–Baxter equation, let $V$ be the representation space, and let a classical dynamical $r$-matrix be a m…

A rack is a set equipped with a binary operation satisfying the rack axioms; two racks are identified when they are isomorphic. Let $n$ denote the size of the rack. Rack enumeratio…

Heckenberger–Lochmann–Vendramin conjecture. Any finite-dimensional elementary Nichols algebra of group type is $bg$-equivalent to one of those listed in Table 9.1 of Heckenberger's…

Yang–Baxter conjecture. The operators $X_i(u)$ obey the Yang–Baxter equation.

Logarithmic multipermutation-level conjecture. One has

Let $\mathfrak{g}=\mathfrak{sl}(n)$, let $\tau$ be a Belavin–Drinfeld triple, and suppose that $r^0=t^0/2+\tilde{r}^0\in\mathfrak{h}\otimes\mathfrak{h}$ is $\tau$-admissible. Defin…

Schneider's conjecture. The constructed solution coincides with the GGS solution.

The twist conjecture. Taking $K_{\alpha,\beta}$ as in the displayed formula, the matrix $R_J$ satisfies the QYBE and coincides with $R_{\text{GGS}}$:

Let $(\Gamma_1,\Gamma_2,\tau)$ be an admissible triple and let $r^0\in\mathfrak h\otimes\mathfrak h$ satisfy the equations referenced in the source. Define $\tilde r^0=r^0-P^0/2$,…

Root-vector factorization conjecture. The following identity holds:

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…

Danilov–Koshevoy's conjecture. The Yang–Baxter equation holds for any crystals $A$, $B$, and $C$.

Let $(X,r)$ be a square-free solution of the set-theoretic Yang–Baxter equation, and let $S(X,r)$ be its associated semigroup. For $x,y,x',y'\in X$, write $r(xy)=y'x'$. Conjectures…

Let $(X,r)$ be a multipermutation square-free solution of level $m$. Generalized twisted-union conjecture. Every multipermutation square-free solution of level $m$ is a generalized…

Let $(X,r)$ be a square-free, nondegenerate, involutive solution of the set-theoretic Yang–Baxter equation, with associated semigroup $S(X,r)$. Skew-polynomial ordering conjecture.…

Let $(X,r)$ be a square-free set-theoretic solution of the Yang–Baxter equation. Retractability conjecture. Every square-free solution of the Yang–Baxter equation is retractable in…

Conjecture 1. A solution to this equation is given in term of an infinite product of explicit invertible elements of $U_q({\mathfrak g})$. This conjecture seeks a universal formula…

Let $b_{CG}$ be the classical boundary solution described in the source, and let $\mathfrak{R}'$ be the set of solutions of the modified quantum Yang–Baxter equation. Let the modif…

Let $V$ be the representation space used in the paper, and let $d4b1'$ denote the set of solutions of the modified quantum Yang–Baxter equation in $\operatorname{End}(V\otimes V)$.…

Let $L$ be a vector space, let $P=\operatorname{id}_{L^{\otimes 2}}+\tau$ and let $I=\operatorname{gen}\{\operatorname{im}P\}$ be the quadratic ideal generated by the image of $P$.…

Let $n$ and $g$ be parameters, and let $\mathcal{G}_{n,g}$ denote the fundamental vector conjugate $A_{n,g}$ graph. Consider solutions of the Yang–Baxter equation based on…

Let $h$ be a local Hamiltonian density, and let the Reshetikhin condition denote the lowest-order nontrivial constraint obtained by expanding the Yang–Baxter equation around a regu…

Extension conjecture. Theorem can be extended to all generalized Belavin–Drinfeld triples.