Exact relation counts and prolongation-sequence periodicity for quadratic algebras of left-nondegenerate idempotent sets (Questions 4.12 and 6.12 of Gateva-Ivanova-Majid)

Gateva-Ivanova and Majid (2024) proved that the quadratic algebra $A(k,X,r)$ of a finite left-nondegenerate idempotent solution of the set-theoretic Yang-Baxter equation is always a PBW algebra of Gelfand-Kirillov dimension 1 with $\dim_k A_d = n = |X|$ for every $d$, and studied its Veronese subalgebras through an infinite prolongation sequence of new solutions $(X, r^{(d)})$ on the same set. The paper explicitly poses two open questions. Question 4.12: for a left-nondegenerate idempotent quadratic set (braid relations not assumed) whose quadratic algebra is PBW with the fixed enumeration and has Gelfand-Kirillov dimension 1, what is the exact upper bound for $\dim_k A_2$, equivalently the minimal possible number of quadratic relations? Only the bounds $n \le \dim_k A_2 \le \binom{n}{2}+1$ are known. Question 6.12: can a braided set be characterised by properties of its prolongation sequence, and in the left-nondegenerate idempotent case what is the stopping distance (the least $d-1$ with $(X,r^{(d)})=(X,r)$) and how many distinct solutions arise?

Background

The Yang-Baxter equation for a linear operator $R:V\otimes V\to V\otimes V$ underlies quantum groups and braid representations; its set-theoretic analogue was proposed in the early 1990s and systematically studied from Etingof-Schedler-Soloviev onwards. A braided set (or set-theoretic solution of the Yang-Baxter equation, YBE) is a pair $(X,r)$ with a map $r:X\times X\to X\times X$ satisfying the braid relations $r^{12}r^{23}r^{12}=r^{23}r^{12}r^{23}$ on $X^3$, where $r^{12}=r\times\mathrm{id}_X$ and $r^{23}=\mathrm{id}_X\times r$. Writing $r(x,y)=({}^{x}y, x^{y})$, the set is left-nondegenerate when $L_x:y\mapsto{}^{x}y$ is bijective for each $x$, involutive when $r^2=\mathrm{id}$, and idempotent when $r^2=r$. The involutive nondegenerate case is algebraically well understood, while the strictly braided (non-involutive) case is far less developed; idempotent solutions form a non-involutive class studied for its own sake.

Gateva-Ivanova and Majid work with the more general notion of a quadratic set: any pair $(X,r)$ with $r:X\times X\to X\times X$, no braid relations or bijectivity assumed. To a finite quadratic set and a field $k$ one associates a quadratic algebra (an algebra generated in degree 1 with only degree-2 relations) $A(k,X,r)=k\langle X\rangle/(\Re_A)$: fixing an enumeration $X=\{x_1,\dots,x_n\}$ and the degree-lexicographic order, the relations rewrite each word $xy\in X^2$ to a degree-lexicographically smaller word of its orbit under the action of $r$ on $X^2$, so $r$ encodes a canonically presented monoid algebra. When $(X,r)$ solves the YBE this is called its Yang-Baxter algebra, a set-theoretic analogue of the quantum plane. A quadratic algebra is PBW (Poincare-Birkhoff-Witt type) when its relations extend to a quadratic Grobner basis, so that the normal words (those avoiding the leading monomials of the Grobner basis) form a $k$-basis; its Gelfand-Kirillov dimension (GKdim) measures the growth of the dimensions of its graded components, with GKdim 1 meaning linear growth.

The 2024 source paper proves that for every finite left-nondegenerate idempotent solution of the YBE the relations $\Re_A$ are themselves the reduced Grobner basis, hence $A(k,X,r)$ is PBW with normal words $\{x_1^{d-1}x_j\}$ and $\dim_k A_d=n$ for all $d\ge 1$ (Hilbert series $(1+(n-1)t)/(1-t)$, GKdim 1), and combines this with a new graph-theoretic theorem on general PBW algebras to deduce infinite global dimension for these algebras. The combinatorics is carried by the directed graph $\Gamma_N$ whose arrows are the length-2 normal words, so $\dim_k A_2$ counts arrows; under GKdim 1 the non-loop part of this graph is a subgraph of an acyclic tournament (an orientation of the complete graph on the $n$ vertices containing no directed cycle) with at most one self-arrow, giving $\dim_k A_2\le\binom{n}{2}+1$, while left-nondegeneracy and idempotence force $\dim_k A_2\ge n$. Crucially, these bounds concern quadratic sets that need not satisfy the braid relations: for actual YBE solutions the value is pinned at $n$, and a quadratic monomial algebra on variables $y_1,\dots,y_n$ whose length-2 normal words are $\{y_iy_j\mid i<j\}\cup\{y_1y_1\}$ attains $\binom{n}{2}+1$, but whether the upper bound or any intermediate value is attained by the algebra of a left-nondegenerate idempotent quadratic set is unknown.

The paper also develops Veronese theory: the $d$-Veronese subalgebra $A^{(d)}$ of a graded algebra is the subalgebra spanned by degrees divisible by $d$, the classical projective-geometry construction whose Yang-Baxter-algebra analogue was built earlier for involutive solutions. In the left-nondegenerate idempotent case the authors show that the set $N_d$ of normal words of degree $d$ carries a canonical left-nondegenerate idempotent solution $(N_d,\rho_d)$ with $N_d\cong X$, and that $A(k,X,r)^{(d)}\cong A(k,X,r^{(d)})$ for a new solution $(X,r^{(d)})$ on the same underlying set; iterating gives the prolongation sequence of $(X,r)$. For the permutation idempotent solutions $r_f(x,y)=(f(y),y)$ studied in the precursor work of the same authors, the sequence is periodic with period the order of $f$ (indeed $A^{(d)}\cong A(k,X,r_{f^d})$), and the paper gives a 3-point example whose prolongation sequence alternates between $r$ and $r^{(2)}$; no general description of the periodicity, the return to $r$, or the number of distinct solutions in the sequence is known.

Problem Statement

The problem consists of the two open questions explicitly posed by Gateva-Ivanova and Majid in arXiv:2409.02939, which share one objective: replace, by exact results, the bounds and partial descriptions that paper establishes for left-nondegenerate idempotent sets and their quadratic algebras. Both deliverables are read off the same normal-word combinatorics of $A(k,X,r)$.

Question 4.12 (exact relation count). Let $(X,r)$ be a left-nondegenerate idempotent quadratic set on $X=\{x_1,\dots,x_n\}$ (that is, $r^2=r$ and each $L_x:y\mapsto{}^{x}y$ is bijective; the braid relations are not assumed). Suppose the associated quadratic algebra $A=A(k,X,r)$ is PBW with PBW-generators $x_1,\dots,x_n$ in this fixed enumeration, and that $\mathrm{GKdim}\,A=1$. Determine the exact upper bound for $\dim_k A_2$, equivalently the minimal possible number of quadratic relations of $A$, as a function of $n$. The known bounds are $n\le\dim_k A_2\le\binom{n}{2}+1$; the lower bound is attained (every left-nondegenerate idempotent solution of the YBE has $\dim_k A_2=n$), and it is open whether the upper bound, or any intermediate value, is attained within the stated class.

Question 6.12 (prolongation dynamics). (1) Can a braided set $(X,r)$ be characterised by properties of its prolongation sequence $(X,r)^{(d)}=(N_d,\rho_d)$, that is, is there a necessary-and-sufficient condition on the prolongation sequence for $(X,r)$ to satisfy the braid relations? For a general braided set the underlying sets in general vary with $d$; in the left-nondegenerate idempotent case the prolongations refer uniformly back to the same set $X$, giving solutions $(X,r^{(d)})$. (2) In the left-nondegenerate idempotent case, what is the stopping distance, defined as $d-1$ for the smallest $d$ such that the $d$-prolongation $(X,r^{(d)})$ returns to $(X,r)$, and how many distinct solutions arise in the sequence? For permutation idempotent solutions $(X,r_f)$ the sequence is periodic with period the order of $f$, and one 3-point example alternates between $r$ and $r^{(2)}$; a general description of the periodicity and of the set of distinct solutions occurring is not known.

A complete answer identifies the exact maximal value of $\dim_k A_2$ (with proof of the universal upper bound over the stated class and explicit attaining examples) and establishes the periodicity structure of prolongation sequences for left-nondegenerate idempotent solutions (stopping distance and number of distinct solutions), settling the characterization part of Question 6.12 either by a theorem or by explicit counterexamples.

References

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