Feigin and Odesskii's elliptic algebras (Q2029196)

\textit{E. K. Sklyanin} [Funct. Anal. Appl. 16, 263--270 (1983; Zbl 0513.58028); translation from Funkts. Anal. Prilozh. 16, No. 4, 27--34 (1982)] defined a family of algebras in the context of elliptic solutions of the Yang-Baxter equation. These so-called Sklyanin algebras algebras have fueled the development of noncommutative projective algebraic geometry, see [\textit{M. Artin} et al., Prog. Math. None, 33--85 (1990; Zbl 0744.14024)]. The properties of these algebras are difficult to discern algebraically, but become apparent when related to the associated twisted homogeneous coordinate ring of an elliptic curve. Subsequently, Sklyanin's definition was generated by \textit{A. V. Odesskiĭ} and \textit{B. L. Feigin} [Funkts. Anal. Prilozh. 23, No. 3, 45--54 (1989; Zbl 0687.17001)]. Fix a point \(\eta \in \mathbb{C}\) lying in the upper half-place, let \(\Lambda=\mathbb{Z} + \mathbb{Z}\eta\), and let \(E=\mathbb{C}/\Lambda\). For positive integers \(n,k\) with \(k<n\) and \(\tau \in \mathbb{C}-\frac{1}{n}\Lambda\), the algebra \(Q_{n,k}(E,\tau)\) is defined as the free algebra \(\mathbb{C}\langle x_0,\dots,x_{n-1}\rangle\) modulo \(n^2\) quadratic relations depending on certain theta functions of order \(n\). The algebras \(Q_{n,n-1}(E,\tau)\) and \(Q_{n,k}(E,0)\) are polynomial rings, facts proved in the present paper. The cases \((n,k)=(3,1)\) and \((4,1)\) are the classical three- and four-dimensional Sklynain algebras, respectively, and are well-understood. The case of \(Q_{n,1}(E,\tau)\) was studied by \textit{J. Tate} and \textit{M. Van den Bergh} [Invent. Math. 124, No. 1--3, 619--647 (1996; Zbl 0876.17010)]. The present paper initializes a project to study the algebras \(Q_{n,k}(E,\tau)\) cohesively. This work provides necessary prerequisite material for material to appear in subsequent papers. In particular, considerable background on theta functions is provided. The authors give various realizations of these algebras. They prove that \(Q_{n,k}(E,\tau) \cong Q_{n,k'}(E,\tau)\) where \(n > k' \geq 1\) and \(kk'=1 \mod n\), a result stated without proof by Feigin and Odesskii, and they show that \(Q_{n,k}(E,\tau) \cong Q_{n,k}(E,-\tau)\). On the other hand, for \(\zeta \in E[n]\), the \(n\)-torsion subgroup of \(E\), \(Q_{n,k}(E,\tau)\) and \(Q_{n,k}(E,\tau+\zeta)\) are graded twists of one another, and hence have equivalent categories of graded left modules. This twist is related to an action by the Heisenberg group \(H_n\) which acts as degree-preserving algebra automorphisms of \(Q_{n,k}(E,\tau)\).