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Difference Operators of Sklyanin and van Diejen Type

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  • Mathematics


The Sklyanin algebra Sη has a well-known family of infinite-dimensional representations D(μ),μ∈C∗ , in terms of difference operators with shift η acting on even meromorphic functions. We show that for generic η the coefficients of these operators have solely simple poles, with linear residue relations depending on their locations. More generally, we obtain explicit necessary and sufficient conditions on a difference operator for it to belong to D(μ) . By definition, the even part of D(μ) is generated by twofold products of the Sklyanin generators. We prove that any sum of the latter products yields a difference operator of van Diejen type. We also obtain kernel identities for the Sklyanin generators. They give rise to order-reversing involutive automorphisms of D(μ) , and are shown to entail previously known kernel identities for the van Diejen operators. Moreover, for special μ they yield novel finite-dimensional representations of Sη .

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