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

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

1998-01-22

arxiv.org/abs/solv-int/9801022v2

Physics

High Energy Physics

High Energy Physics - Theory

27 pages, latex, typos corrected

Scientific paper

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.

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