Vacuum curves of elliptic L-operators and representations of Sklyanin algebra

Abstract

An algebro-geometric approach to representations of Sklyanin algebra is proposed. To each 2 \times 2 quantum L-operator an algebraic curve parametrizing its possible vacuum states is associated. This curve is called the vacuum curve of the L-operator. An explicit description of the vacuum curve for quantum L-operators of the integrable spin chain of XYZ type with arbitrary spin $\ell$ is given. The curve is highly reducible. For half-integer $\ell$ it splits into $\ell +1/2$ components isomorphic to an elliptic curve. For integer $\ell$ it splits into $\ell$ elliptic components and one rational component. The action of elements of the L-operator to functions on the vacuum curve leads to a new realization of the Sklyanin algebra by difference operators in two variables restricted to an invariant functional subspace.

27 pages, latex, typos corrected

References in corpus (2)

• Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations
• Spin generalization of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra

Cited by in corpus (5)

• Yang-Baxter R operators and parameter permutations
• Baxter Q-operators of the XXZ chain and R-matrix factorization
• New elliptic solutions of the Yang-Baxter equation
• Intertwining operators for Sklyanin algebra and elliptic hypergeometric series
• Matrix factorization for solutions of the Yang-Baxter equation