The 4-dimensional Sklyanin algebra
Let \(A = T(V)/ \langle R\rangle\) be the 4-dimensional Sklyanin algebra attached to an elliptic curve \(E\) and a point \(\tau \in E\) not of order 4; recall that \(V\) is the space of sections of a certain line bundle \(L\) of degree 4 over \(E\) and \(R\subset V\otimes V\). \(A\) has two central elements of degree 2; let \(\Omega \subset V \otimes V\) be the vector space spanned by these elements. Now \(V\) can be identified with the space generated by the 4 Jacobi theta functions, and hence \(V \otimes V\) can be thought of as a space of functions on \(\mathbb{C} \times \mathbb{C}\). The main results of this paper present functional equations satisfied by, and determining, the elements of the spaces \(R\) and \(\Omega\). This allows for new proofs of results from [\textit{E. K. Sklyanin}, Funct. Anal. Appl. 17, 273-284 (1983); translation from Funkts. Anal. Prilozh. 17, No. 4, 34-48 (1983; Zbl 0536.58007)] and [\textit{T. Levasseur} and \textit{S. P. Smith}, Bull. Soc. Math. Fr. 121, 35-90 (1993)], concerning respectively the action of \(A\) on the space of meromorphic functions on one variable, and the annihilators of the point modules.
scientific article; zbMATH DE number 614201