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Difference Operators of Sklyanin and van Diejen Type

Rains, Eric and Ruijsenaars, Simon (2013) Difference Operators of Sklyanin and van Diejen Type. Communications in Mathematical Physics, 320 (3). pp. 851-889. ISSN 0010-3616. doi:10.1007/s00220-013-1692-3.

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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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Additional Information:© 2013 Springer-Verlag Berlin Heidelberg. Received: 6 March 2012; Accepted: 6 October 2012; Published online: 5 March 2013. This collaboration was begun while the authors were visiting the Liu Bie Ju Centre at City University of Hong Kong. We would like to thank the Centre and M. Ismail for the invitation, hospitality and financial support. The first author was supported in part by a grant (DMS-1001645) from the National Science Foundation. Finally, thanks are due to the referee for a careful report, which helped to improve the exposition.
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Issue or Number:3
Record Number:CaltechAUTHORS:20130723-114503880
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Official Citation:Rains, E. & Ruijsenaars, S. Commun. Math. Phys. (2013) 320: 851.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:39526
Deposited On:24 Jul 2013 21:04
Last Modified:09 Nov 2021 23:45

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