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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. https://resolver.caltech.edu/CaltechAUTHORS:20130723-114503880

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Abstract

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 .


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00220-013-1692-3DOIArticle
http://link.springer.com/article/10.1007%2Fs00220-013-1692-3PublisherArticle
http://rdcu.be/v7gOPublisherFree ReadCube access
https://arxiv.org/abs/1203.0042arXivDiscussion Paper
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.
Funders:
Funding AgencyGrant Number
NSFDMS-1001645
Issue or Number:3
DOI:10.1007/s00220-013-1692-3
Record Number:CaltechAUTHORS:20130723-114503880
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20130723-114503880
Official Citation:Rains, E. & Ruijsenaars, S. Commun. Math. Phys. (2013) 320: 851. https://doi.org/10.1007/s00220-013-1692-3
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:39526
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:24 Jul 2013 21:04
Last Modified:09 Nov 2021 23:45

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