23 problems

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…

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 $\tilde r^0=r^0-P^0/2$,…

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 $\mathfrak h$ 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…

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…

A set-theoretic solution of the Yang–Baxter equation is a solution considered up to isomorphism, and its size is the cardinality of its underlying set. Let $n$ denote the size of t…

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…

Rectangular multi-$t$ Yang–Baxter trace conjecture. The equivariant trace of $Y_{c3_bc}(v_bc)$ is proportional to

Tableau-process weight conjecture. This weight is proportional to the weight derived from the tableau process of Ayyer et al.

Let $(X,r)$ be an involutive solution of size $n$, and let $g(n)$ denote the bound used in the surrounding text for solutions whose permutation group is abelian. Abelian permutatio…

Let $(X,r)$ be an involutive solution of size $n$, and define … Dehornoy's class bound conjecture. The Dehornoy's class is bounded by $a_n$, and the bound $a_n$ is minimal. The cla…

Let $(X,r)$ be an indecomposable involutive solution, and let the Dehornoy's class denote its associated class invariant. Dehornoy's class conjecture for indecomposable solutions.…

Let $X$ be an indecomposable solution of the Yang–Baxter equation whose permutation group is isomorphic to the dihedral group $D_{2n}$. Dihedral size conjecture. Then … For an inde…

Orbit-profile invariance conjecture. Equality of the orbit profiles should preserve the stated dimension, or Gelfand–Kirillov dimension, under the conditions intended in the source…

A Rump right quasigroup is a set equipped with the right-quasigroup structure arising in Rump's framework for set-theoretic solutions of the Yang–Baxter equation. Its set-theoretic…

Let $V$ be a finite-dimensional complex vector space, and let $R$ be a unitary solution to the Yang–Baxter equation on $V^{\otimes 2}$. Suppose that $R$ has finite projective order…

Let $X$ be a set with an involutive solution $csigma$ of the Yang–Baxter equation, satisfying $csigma(x,y)^2=(x,y)$. Let $B$ be the differential bialgebra associated to this soluti…

Gateva-Ivanova's conjecture. Every finite nondegenerate involutive square-free set-theoretic solution $r:X\times X\to X\times X$ such that $|X|\geq2$ is a multipermutation solution…

Let $G$ be an affine connected nilpotent algebraic group over $\mathbb{C}$, and let $\mathfrak{g}$ be its Lie algebra. A Hopf 2-cocycle on the coordinate Hopf algebra of $G$ is con…

Gekhtman–Shapiro–Vainshtein correspondence conjecture. Let $\mathcal{G}$ be a simple complex Lie group. For any Belavin–Drinfeld triple $T=(\Gamma_{1},\Gamma_{2},\gamma)$ there exi…

Let $X$ be a simple object in a fusion category $\mathcal{C}$ equipped with a Yang–Baxter operator $(X,c)$. Let $(YB_{(X,c)},\rho^{(X,c)},\iota)$ denote the associated sequence of…

Logarithmic multipermutation-level conjecture. One has

Retractability and multipermutation conjecture. Every finite square-free solution $(X,r)$ is retractable. Moreover, every finite square-free solution $(X,r)$ of finite order $n$ is…