On the cluster nature and quantization of geometric R-matrices

Rei Inoue, Thomas Lam, Pavlo Pylyavskyy

Research output: Contribution to journalArticlepeer-review

Abstract

We define cluster R-matrices as sequences of mutations in triangular grid quivers on a cylinder, and show that the affine geometric R-matrix of symmetric power representations for the quantum affine algebra U’ q (ŝl n ) can be obtained from our cluster R-matrix. A quantization of the affine geometric R-matrix is defined, compatible with the cluster structure. We construct invariants of the quantum affine geometric R-matrix as quantum loop symmetric functions.

Original languageEnglish (US)
Pages (from-to)25-78
Number of pages54
JournalPublications of the Research Institute for Mathematical Sciences
Volume55
Issue number1
DOIs
StatePublished - 2019

Bibliographical note

Funding Information:
We thank the anonymous referees for helpful comments. R.I. was partially supported by JSPS KAKENHI grants 26400037 and 16H03927. T.L. was partially supported by NSF grants DMS-1160726, DMS-1464693, and a Simons Fellowship. P.P. was partially supported by NSF grants DMS-1148634, DMS-1351590, and a Sloan Fellowship.

Publisher Copyright:
© 2019 Research Institute for Mathematical Sciences, Kyoto University.

Keywords

  • Cluster algebra
  • Geometric r-matrix
  • Quantum cluster algebra

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