Isomorphism classification of three-generator Sklyanin algebras over fields of characteristic 3 and over fields without a nontrivial cubic root of unity

The Sklyanin algebra $Q_{p,q,r}$ over a field $\mathbb{K}$ is the quadratic algebra on three generators $x,y,z$ with the relations $pyz+qzy+rxx=0$, $pzx+qxz+ryy=0$, $pxy+qyx+rzz=0$. Iyudu and Shkarin (arXiv:2108.06290) determined exactly which of these algebras are isomorphic, provided the ground field has characteristic different from 3 and contains a nontrivial cubic root of unity $\theta$: the answer is given by the orbits of a 24-element group isomorphic to $SL_2(\mathbb{Z}_3)$ (the group of $2\times 2$ matrices of determinant 1 over the three-element field) acting by fractional transformations on the normalized parameter surface, together with explicitly described degenerate and quantum-polynomial strata. The authors state explicitly that they did not treat the two remaining regimes: fields of characteristic 3, and fields of characteristic different from 3 containing no nontrivial cubic root of unity (such as the rationals, the reals, or $\mathbb{F}_p$ with $p\equiv 2 \pmod{3}$). The present problem asks for the explicit parameter-space classification of the isomorphism classes in those two regimes. Both are genuinely different from the solved case: in characteristic 3 the root-of-unity substitution collapses and the degeneracy stratification itself changes shape ($p^3=q^3=r^3$ reduces to $p=q=r$), while over fields without $\theta$ the classification over the algebraic closure does not automatically descend, raising the question of whether distinct $\mathbb{K}$-forms of Sklyanin algebras — algebras that become isomorphic only after extending the ground field — exist.

Background

The three-generator Sklyanin algebras form one of the most studied families of graded algebras in noncommutative algebra. Recall that a quadratic algebra is a graded algebra $A(V,R)$ presented by a finite-dimensional vector space $V$ of degree-one generators and a subspace $R\subseteq V\otimes V$ of homogeneous quadratic relations; its Hilbert series $H_A(t)=\sum_{n\ge 0}\dim A_n\,t^n$ is its basic size invariant. The Sklyanin algebra $Q_{p,q,r}$ over a field $\mathbb{K}$, introduced by E. K. Sklyanin in connection with the Yang–Baxter equation (the consistency condition for the factorized scattering operators of exactly solvable models in statistical mechanics and quantum field theory), is the quadratic algebra on generators $x,y,z$ with the three relations $pyz+qzy+rxx=0$, $pzx+qxz+ryy=0$, $pxy+qyx+rzz=0$, where $p,q,r\in\mathbb{K}$. Multiplying the triple $(p,q,r)$ by a common nonzero scalar does not change the algebra, so the parameters are naturally points of a projective plane; nevertheless, non-proportional triples can give isomorphic algebras, and describing exactly when this happens is a delicate question.

The landmark results are geometric. Artin and Schelter classified graded algebras of global dimension 3 that are Artin–Schelter regular (the noncommutative analogues of polynomial rings); Artin, Tate and Van den Bergh then showed, over an algebraically closed ground field, that $Q_{p,q,r}$ with at least two of $p,q,r$ nonzero and $p^3,q^3,r^3$ not all equal is Artin–Schelter regular: it is Koszul (its trivial module admits a free resolution generated in matching degrees) and has the same Hilbert series $(1-t)^{-3}$ as a commutative polynomial ring in three variables (a PHS-algebra, for polynomial Hilbert series). Their method attaches to $Q_{p,q,r}$ geometric data — a smooth elliptic curve $E\subset\mathbb{P}^2$ parametrizing the point modules of the algebra (a point module is a cyclic graded module all of whose graded components are one-dimensional — the noncommutative analogue of a point of $\mathbb{P}^2$) together with a translation automorphism — and shows two Sklyanin algebras are isomorphic precisely when their geometric data agree, a criterion that is not explicit in the parameters $(p,q,r)$. Odesskii and Feigin placed the family inside the wider theory of elliptic algebras; Polishchuk and Positselski proved that a generic $Q_{p,q,r}$ over $\mathbb{C}$ is Koszul but not a PBW-algebra, i.e. no choice of bases of generators and relations turns the relations into a noncommutative Gröbner basis with respect to a monomial ordering (the noncommutative analogue of a Poincaré–Birkhoff–Witt basis); Smith showed that the degenerate Sklyanin algebras, those failing the regularity criterion, become monomial after a linear change of generators; Walton determined the dimensions of the irreducible representations of the regular members of the family and their polynomial-identity degree in the module-finite case. All of these treatments work over algebraically closed fields, in practice of characteristic other than 2 or 3.

Iyudu and Shkarin re-established and extended these facts by purely algebraic, combinatorial means. Their 2017 paper constructed Gröbner bases for Sklyanin algebras indirectly, through a one-sided module; the 2021 paper, the source of the present problem, found a linear substitution after which the leading monomials of a reduced Gröbner basis of the ideal of relations of $Q_{p,q,1}$ itself can be read off. The substitution uses a nontrivial cubic root of unity, an element $\theta\in\mathbb{K}$ with $\theta^3=1\neq\theta$ (which exists in every algebraically closed field of characteristic different from 3 and in no field of characteristic 3). After scaling and a chain of substitutions the algebra is brought to the potential algebra with potential $x^3-(xyz)^{\mathrm{cyc}}+\alpha(yyz)^{\mathrm{cyc}}-(\gamma-\alpha^2)z^3$, where $u^{\mathrm{cyc}}$ denotes the cyclic symmetrization of a word $u$ (the sum of all cyclic shifts of $u$) and a potential algebra is one whose relations are the formal partial derivatives of such a cyclically invariant potential; an induction then shows the Gröbner basis is either an explicitly described infinite family with a regular pattern or a finite one. This yields a short proof of the Koszulity and Hilbert-series theorem (valid over every field of characteristic different from 3, with or without $\theta$ in it, since Hilbert series and Koszulity are unchanged by extending the ground field within the same characteristic): $Q_{p,q,r}$ is always Koszul, and it is a PHS-algebra if and only if at least two of $p,q,r$ are nonzero and $p^3=q^3=r^3$ fails.

The same paper then settles the isomorphism problem in that setting. First, a general observation valid over any field: if two quadratic algebras are isomorphic as algebras they are already isomorphic as graded algebras, that is, by a linear substitution of the generators, so 'isomorphic' is unambiguous. Assuming $\theta\in\mathbb{K}$ and characteristic different from 3, the classification reads: the degenerate algebras reduce to the free algebra and two monomial algebras (with relations $xy=zx=yz=0$ and $xx=yy=zz=0$); the intermediate stratum — the parameters with $r=0$ and $pq\neq 0$, together with those with $(p+q)^3+r^3=0$ for which $p^3=q^3=r^3$ fails — consists of quantum polynomial algebras $B_\alpha$ with relations $xy=\alpha yx$, $zx=\alpha xz$, $yz=\alpha zy$, where $B_\alpha\cong B_\beta$ if and only if $\alpha=\beta$ or $\alpha\beta=1$; and on the remaining main stratum $r\neq 0$, $(p,q)\neq(0,0)$, $(p+q)^3+r^3\neq 0$, normalized to $r=1$, two algebras $Q_{a,b,1}$ and $Q_{a',b',1}$ are isomorphic exactly when $(a,b)$ and $(a',b')$ lie in the same orbit of a 24-element group of fractional transformations isomorphic to $SL_2(\mathbb{Z}_3)$, the group of $2\times 2$ matrices of determinant 1 over the three-element field, generated by $(a,b)\mapsto(\theta a,\theta b)$ and $(a,b)\mapsto\bigl((\theta a+\theta^2 b+1)/(a+b+1),(\theta^2 a+\theta b+1)/(a+b+1)\bigr)$.

Every ingredient of this description breaks in the two regimes the authors explicitly set aside. In characteristic 3 a nontrivial cubic root of unity cannot exist (there $t^3-1=(t-1)^3$), so neither the substitution nor the orbit formulas even make sense; moreover the parameter-space geometry itself changes, because the Frobenius map $u\mapsto u^3$ is injective in characteristic 3, so the degeneracy equation $p^3=q^3=r^3$ collapses to $p=q=r$ and $x^3+y^3+z^3=(x+y+z)^3$, whence the degenerate strata and the elliptic-curve data behave differently from all other characteristics. In characteristic different from 3 but without $\theta$ (for instance $\mathbb{Q}$, $\mathbb{R}$, or $\mathbb{F}_p$ with $p\equiv 2 \pmod{3}$), the Hilbert-series and Koszulity statements survive by passing to the algebraic closure, but isomorphism does not: two algebras with $\mathbb{K}$-rational parameters may be joined by a linear substitution only over the quadratic extension $\mathbb{K}(\theta)$, and whether a substitution over $\mathbb{K}$ itself must exist — a Galois-descent question, with room for genuine twist phenomena — distinct $\mathbb{K}$-forms, that is, algebras that become isomorphic only after extending the ground field — at special parameters whose automorphism group is larger than the generic one, is not known. The open problem is to complete the classification in precisely these two regimes.

Problem Statement

Let $\mathbb{K}$ be a field. The three-generator Sklyanin algebra with parameters $(p,q,r)\in\mathbb{K}^3$ is the quadratic algebra $Q_{p,q,r}$ over $\mathbb{K}$ on generators $x,y,z$ with the three defining relations $pyz+qzy+rxx=0$, $pzx+qxz+ryy=0$, $pxy+qyx+rzz=0$; multiplying the triple $(p,q,r)$ by a common nonzero scalar does not change the algebra. Determine, explicitly in terms of the parameter triples, necessary and sufficient conditions for $Q_{p,q,r}$ and $Q_{p',q',r'}$ to be isomorphic as $\mathbb{K}$-algebras (equivalently, by a general observation of Iyudu–Shkarin valid for quadratic algebras over any field, as graded algebras via a linear substitution of the generators), in each of the two regimes not covered by the existing classification:

(i) $\mathrm{char}\,\mathbb{K}=3$;

(ii) $\mathrm{char}\,\mathbb{K}\neq 3$ and $\mathbb{K}$ contains no element $\theta$ with $\theta^3=1\neq\theta$ (for example $\mathbb{Q}$, $\mathbb{R}$, or $\mathbb{F}_p$ with $p\equiv 2 \pmod{3}$).

A complete solution should, in each regime: identify the degenerate strata of the parameter space — the loci where the algebra ceases to have the polynomial Hilbert series $(1-t)^{-3}$, which for characteristic different from 3 are exactly the triples with at least two of $p,q,r$ zero or with $p^3=q^3=r^3$ (in characteristic 3 the locus $p^3=q^3=r^3$ collapses to the diagonal $p=q=r$, so this stratification must itself be re-derived); treat any intermediate stratum analogous to the quantum-polynomial family (algebras with relations of the shape $xy=\alpha yx$, $zx=\alpha xz$, $yz=\alpha zy$); give the orbit or group-theoretic description of the main stratum over $\mathbb{K}$; and, in regime (ii), settle the descent question of whether two $\mathbb{K}$-rational parameter triples lying in a common orbit over the algebraic closure are always joined by an isomorphism defined over $\mathbb{K}$, or whether genuinely distinct $\mathbb{K}$-forms occur — algebras that become isomorphic only after extending the ground field — in particular at special parameters whose stabilizer or automorphism group is nontrivial.

References

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