Ruijsenaars' commuting difference operators as commuting transfer matrices

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58 Citations (Scopus)


For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set of difference operators acting on functions on the type A weight space. According to the fundamental relation RLL = LLR, taking the trace of the L-operator gives a set of commuting difference operators. We show that for the above mentioned L-operator this approach gives Macdonald type operators with elliptic theta function coefficient, actually equivalent to Ruijsenaars' operators. The relationship between the difference L-operator and Krichever's Lax matrix is given, and an explicit formula for elliptic commuting differential operators is derived. We also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space.

Original languageEnglish
Pages (from-to)289-325
Number of pages37
JournalCommunications in Mathematical Physics
Issue number2
Publication statusPublished - 1997 Aug 1

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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