The transitive-array conjecture for solutions of the classical Yang–Baxter equation

From papers

Let $C$ be a set, let $\boldsymbol{a}$ be a transitive $n\times n$ matrix with entries in $C$ , and let $\mathbf r=\{r^{(c)}\}_{c\in C}\subset \mathfrak{g}\otimes\mathfrak{g}$ be a solution of the transitive classical Yang–Baxter equation. Write $\mathbf r^{(\boldsymbol{a})}$ for the corresponding element constructed from $\boldsymbol{a}$ and $\mathbf r$ . Transitive-array conjecture. For every such $\boldsymbol{a}$ and $\mathbf r$ , the element $\mathbf r^{(\boldsymbol{a})}$ solves the classical Yang–Baxter equation. This is a natural generalization of the preceding conjecture and is verified for $n\leq 4$ ; the assertion remains open in general.

Progress summary

Open

The conjecture remains unproved in general: it works in small dimensions and in one special case, but no broader proof or counterexample has been reported.

The conjecture says that every transitive array and every compatible transitive Yang–Baxter family produces a solution of the classical Yang–Baxter equation. Its general validity is recorded as open.

Known results

Verified for $n\leq 4$ .
The preceding binary-valued conjecture is also verified for $n\leq 4$ .
The almost skew-symmetric transitive case is proved.

February 2026 restatement

Berenstein, Greenstein, and Li’s paper Monomial bialgebras states the conjecture as Conjecture 1.5 and reports no general proof, counterexample, or subsequent verification.

Current status (as of August 2026): The conjecture is settled only for $n\leq 4$ and the almost skew-symmetric special case; the general transitive-array case remains open.

Sources

Sources & referencesView supporting material

Primary source

Arkady Berenstein, Jacob Greenstein and Jian-Rong Li, “Monomial bialgebras”, arXiv:2602.02342 (2026).

1 week ago

Let g be any Lie algebra, C any index set, and {r^(c):c∈C}⊂g⊗g a family satisfying the source's transitive classical Yang–Baxter equation: [r^(c)_12,r^(c')_13]+[r^(c)_12,r^(c'')_23]+[r^(c')_13,r^(c'')_23]=0 whenever c'∈{c,c''}.

Let a=(a_ij) be any transitive n×n array, so a_ik∈{a_ij,a_jk} for every i,j,k. Writing ι_u:g→g^{⊕n} for inclusion into the uth direct summand, the source's constructed tensor is R=Σ_{u,v=1}^n (ι_u⊗ι_v)r^(a_{v,u}).

Distinct direct summands of g^{⊕n} commute. Therefore, for every triple (u,v,w), the component of CYB(R)=[R_12,R_13]+[R_12,R_23]+[R_13,R_23] in ι_u(g)⊗ι_v(g)⊗ι_w(g) is exactly [r^(a_{v,u})12,r^(a{w,u})13] +[r^(a{v,u})12,r^(a{w,v})23] +[r^(a{w,u})13,r^(a{w,v})23]. Taking (i,j,k)=(w,v,u) in array transitivity gives a{w,u}∈{a_{w,v},a_{v,u}}. Hence the displayed component vanishes by the transitive classical Yang–Baxter equation with c=a_{v,u}, c'=a_{w,u}, c''=a_{w,v}.

The argument applies without change when indices coincide. Since the tensor cube of g^{⊕n} is the direct sum of all such (u,v,w)-components, every component of CYB(R) vanishes. Consequently CYB(R)=0 for every n, every transitive array, every index set C, and every transitive Yang–Baxter family, proving the full conjecture.

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Shivam Patel · 21 hours ago