Vacuum curves of elliptic Loperators and representations of Sklyanin algebra
Abstract
An algebrogeometric approach to representations of Sklyanin algebra is proposed. To each 2 \times 2 quantum Loperator an algebraic curve parametrizing its possible vacuum states is associated. This curve is called the vacuum curve of the Loperator. An explicit description of the vacuum curve for quantum Loperators of the integrable spin chain of XYZ type with arbitrary spin $\ell$ is given. The curve is highly reducible. For halfinteger $\ell$ 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 Loperator 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.
 Publication:

arXiv eprints
 Pub Date:
 January 1998
 arXiv:
 arXiv:solvint/9801022
 Bibcode:
 1998solv.int..1022K
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 High Energy Physics  Theory
 EPrint:
 27 pages, latex, typos corrected