We explicitly construct two classes of infinitly many commutative operators in terms of the deformed Virasoro algebra. We call one of them local integrals and the other nonlocal one, since they can be regarded as elliptic deformations of the local and nonlocal integrals of motion obtained by V.Bazhanov, S.Lukyanov and Al.Zamolodchikov [1].

We explicitly construct two classes of infinitly many commutative operators in terms of the deformed Virasoro algebra. We call one of them local integrals and the other nonlocal one, since they can be regarded as elliptic deformations of the local and nonlocal integrals of motion obtained by V.Bazhanov, S.Lukyanov and Al.Zamolodchikov [1].

We develop a theory for the Hermite-Krichever Ansatz on the Heun equation. As a byproduct, we find formulae which reduce hyperelliptic integrals to elliptic ones.

The Sklyanin algebra admits realizations by difference operators acting on theta functions. Sklyanin found an invariant metric for the action and conjectured an explicit formula for the corresponding reproducing kernel. We prove this conjecture, and also give natural biorthogonal and orthogonal bases for the representation space. Moreover, we discuss connections with elliptic hypergeometric ser...

It is well known that any elliptic integral can be transformed into a linear combination of elementary functions and Legendre’s three Elliptic functions. Methods for transforming these integrals to the Legendre form are described in numerous papers and textbooks. However, when it comes to actually designing and implementing such a reduction algorithm the existing methods require significant mod...

We discuss the construction of the spectral curve and the action integrals for the " elliptic " XY Z spin chain of the length Nc. Our analysis can reflect the integrable structure behind the " elliptic " N = 2 supersymmetric QCD with N f = 2Nc .