Maximal level of a good specialization and explicit forms of the dimension-$s$ irreducible representations of a PI 4-dimensional Sklyanin algebra

Two open questions recorded by Walton, Wang and Yakimov in their study of the representation theory of polynomial-identity (PI) 4-dimensional Sklyanin algebras (Selecta Math. 27 (2021), Paper No. 99) remain unanswered: (i) Question 6.11, asking for the maximal level $N$ of a good specialization of such an algebra $S$, for which the source establishes only $N \geq 1$ and the non-explicit upper bound $N < M_{\min}$ of Lemma 6.5; and (ii) Question 8.21, asking for the explicit forms of the irreducible representations of $S$ of dimension $s = n/\gcd(n,2)$ other than the four theta-function modules $V(\omega+(s-1)\tau)$, for which the source gives only existence and count statements (Theorems 8.29 and 8.30). Neither question has been answered in the subsequent literature.

Background

The 4-dimensional Sklyanin algebras were introduced by E. K. Sklyanin in his study of the eight-vertex model and the Yang-Baxter equation. The algebra $S = S(\alpha,\beta,\gamma)$ over a field $\Bbbk$ is the $\mathbb{N}$-graded algebra on four generators $x_0,\dots,x_3$ of degree one subject to six quadratic relations depending on parameters $\alpha,\beta,\gamma \in \Bbbk$ with $\alpha+\beta+\gamma+\alpha\beta\gamma = 0$; for generic parameters it is Artin-Schelter regular — the noncommutative analogue of a polynomial ring in four variables — and is central to noncommutative projective geometry. Its geometry is governed by an elliptic curve $E \subset \mathbb{P}^3$ (the intersection of two quadrics) together with an automorphism $\sigma$ of $E$ given by translation by a point $\tau \in E$: the point scheme of $S$ — the parameter space of its point modules, cyclic graded modules of Hilbert series $(1-t)^{-1}$ that play the role of points in noncommutative projective geometry — is $E$ plus four isolated points, and the point and line modules of $S$ are parametrized by the secant geometry of $E$. When $\tau$ has finite order $n$ (equivalently $|\sigma| = n$), $S$ is module-finite over its center $Z = Z(S)$ and is a polynomial identity (PI) algebra — it satisfies a nontrivial polynomial identity, forcing every irreducible representation to be finite-dimensional — of PI degree $n$, which is a sharp upper bound on the dimensions of the irreducible representations; $n$ plays the role of the quantum characteristic. In this PI case, Smith and Tate proved that $Z$ is generated by four elements $z_0,\dots,z_3$ of degree $n$ and two elements $g_1, g_2$ of degree $2$, subject to two relations $F_1, F_2$ of degree $2n$, so the maximal spectrum $Y = \operatorname{maxSpec} Z$ is a 4-dimensional affine algebraic variety; put $s = n/\gcd(n,2)$, that is, $s = n$ for $n$ odd and $s = n/2$ for $n$ even. Two structures frame the present questions. First, Poisson orders via deformation. A Poisson order in the sense of Brown and Gordon is a triple $(A, C, \partial)$ with $C$ a central subalgebra of $A$ and $\partial: C \to \operatorname{Der}(A/C)$ a Lie homomorphism, which induces a Poisson bracket on $\operatorname{maxSpec} C$ whose symplectic leaves control the representation theory of $A$. Walton-Wang-Yakimov construct a formal deformation $S_\hbar$ of $S$ over $\Bbbk[[\hbar]]$ (the parameters are deformed so that the lifted automorphism $\sigma_\hbar$ has infinite order) together with the specialization map $\theta_S: S_\hbar \twoheadrightarrow S$, reduction modulo $\hbar$. A degree-zero section $\iota: Z \hookrightarrow S_\hbar$ of $\theta_S$ is called good if $\iota(g_j) = \widetilde{g}_j$ for $j = 1,2$ and each $\iota(z_i) - \widetilde{z}_i$ lies in the ideal $(\widetilde{g}_1, \widetilde{g}_2)$ of $\Bbbk[\widetilde{x}_i, \widetilde{g}_1, \widetilde{g}_2, \hbar]$, where tildes denote chosen lifts (Definition 6.3 of the source). A good specialization of level $N \geq 1$ is one admitting a good section with $[\iota(z), y] \in \hbar^N S_\hbar$ for all $z \in Z$, $y \in S_\hbar$; such specializations produce Poisson orders on $S$ whose induced Poisson bracket on $Z$ is nonzero and given by an explicit Jacobian formula (Theorem 6.10 of the source). The levels are bounded above: passing to a formal localization $R_\hbar$, there is a least integer $M_{\min}$ with $r^{\sigma_\hbar^n} - r \notin \hbar^{M_{\min}} R_\hbar$ for suitable $r$, and every level satisfies $N < M_{\min}$ (Lemma 6.5 of the source); the maximal level itself is not determined. Second, irreducible representations. Since $S$ is PI, all its irreducible representations are finite-dimensional. They are studied through fat point modules — cyclic graded modules of Hilbert series $d(1-t)^{-1}$ representing the simple objects of the noncommutative projective quotient category of graded $S$-modules modulo those bounded above. Smith classified them: point modules $M(p)$ for $p \in E$; fat point modules $F(\omega + k\tau)$ of multiplicity $k+1$ for $\omega \in E_2$ (the group of four 2-torsion points of $E$) and $0 \leq k \leq s-1$; all other fat points have multiplicity $s$. Sklyanin constructed, for each $\omega \in E_2$ and $k \geq 0$, a $(k+1)$-dimensional representation $V(\omega + k\tau)$ inside a space of theta functions of order $2k$ on $E$ (classical elliptic theta functions), and Smith and Staniszkis proved these are irreducible for $k < s$. The relation $F(\omega+k\tau) \cong \bigoplus_{i \geq 0} V(\omega+k\tau)t^i$ holds for multiplicity below $s$ and extends to $k = s-1$, yielding four irreducible representations $V(\omega+(s-1)\tau)$ of dimension $s$. Walton-Wang-Yakimov then classify the irreducible representations of $S$ by dimension and by the location of the central annihilator in $Y$ (Theorems 8.29 for $n$ odd and 8.30 for $n$ even); in the even case $n = 2s$, for instance, over each point of $Y^{\operatorname{sing}} \setminus \bigcup_{\omega,k} \mathcal{C}(\omega+k\tau)$ there are exactly two irreducible representations of dimension $s$, which arise as simple quotients of fat point modules of multiplicity $s$, while over the smooth locus of $Y$ — the Azumaya locus, the maximal ideals annihilating irreducible representations of the maximum dimension $n$ — there is exactly one irreducible representation, of dimension $n$. These are existence-and-count statements: no explicit form (matrix action of the generators) is given for the dimension-$s$ irreducible representations other than the four theta modules, and the source notes that the resulting dimension spectrum matches that of the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$ at roots of unity.

Problem Statement

The problem consists of the two questions explicitly recorded as open by Walton-Wang-Yakimov (Selecta Math. 27 (2021)) for a PI 4-dimensional Sklyanin algebra $S = S(\alpha,\beta,\gamma)$ with $|\sigma| = n$ finite (the source's standing hypothesis takes $n \geq 3$ and excludes $n$ dividing 4), center $Z$, maximal spectrum $Y = \operatorname{maxSpec} Z$, and $s = n/\gcd(n,2)$. Part (A) (Question 6.11 of the source). Determine the maximal level $N$ of a good specialization $\theta_S: S_\hbar \twoheadrightarrow S$: the largest integer $N$ for which there exists a good degree-zero section $\iota: Z \hookrightarrow S_\hbar$ in the sense of Definition 6.3 of the source (that is, $\iota(g_j) = \widetilde{g}_j$ for $j = 1,2$ and $\iota(z_i) - \widetilde{z}_i \in (\widetilde{g}_1, \widetilde{g}_2)$ for $0 \leq i \leq 3$) such that $[\iota(z), y] \in \hbar^N S_\hbar$ for all $z \in Z$ and $y \in S_\hbar$. As in the source's Question 6.11, the formal deformation $S_\hbar$ — and hence the choice of formal parameter lifts $(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma})$ in the sense of the source's Notation 5.1 that determines it — is fixed when the question is posed. A complete answer states $N$ — as a specific integer or as a formula in $n$ and the parameters $(\alpha,\beta,\gamma)$ — with a proof that a good specialization of level $N$ exists and a proof that no good specialization of $S$ has level $> N$. The source establishes only $N \geq 1$, the existence of a maximum-level good specialization, and the non-explicit bound $N < M_{\min}$ from Lemma 6.5, and it suggests computing $M_{\min}$ (defined in equation (6.8) of the source via $r^{\sigma_\hbar^n} - r \notin \hbar^{M} R_\hbar$) as a route. Part (B) (Question 8.21 of the source). Apart from the four irreducible Sklyanin theta modules $V(\omega+(s-1)\tau)$, $\omega \in E_2$, and their scalar twists, determine the forms of the remaining irreducible representations of $S$ of dimension $s$. Following Section 8.2 of the source, this part is posed over the complex numbers ($\Bbbk = \mathbb{C}$), the setting in which the theta modules $V(\omega+k\tau)$ and the counting Theorems 8.29 and 8.30 are established. A complete answer provides an explicit construction of each isomorphism class of $s$-dimensional irreducible representation — for example, the action of the generators $x_0,\dots,x_3$ as explicit matrices depending on the central character, or an explicit theta-function model — together with proofs of irreducibility, of the stated dimension, of pairwise non-isomorphism, and of completeness (every $s$-dimensional irreducible representation of $S$ is isomorphic to one on the list), consistent with the counting data of Theorems 8.29 and 8.30 of the source. In the even case $n = 2s$ this includes, in particular, the two $s$-dimensional irreducible representations attached to each point $y \in Y^{\operatorname{sing}} \setminus \bigcup_{\omega,k} \mathcal{C}(\omega+k\tau)$, which Theorem 8.30(3) realizes as simple quotients of fat point modules of multiplicity $s$; in the odd case $s = n$ it includes the generic irreducible representations over the Azumaya locus. A complete answer addresses both Part (A) and Part (B).

References

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