On the cluster nature and quantization of geometric R-matrices

Rei Inoue; Thomas Lam; Pavlo Pylyavskyy

doi:10.4171/PRIMS/55-1-2

On the cluster nature and quantization of geometric R-matrices

Rei Inoue, Thomas Lam, Pavlo Pylyavskyy

Mathematics

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11 Scopus citations

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 language	English (US)
Pages (from-to)	25-78
Number of pages	54
Journal	Publications of the Research Institute for Mathematical Sciences
Volume	55
Issue number	1
DOIs	https://doi.org/10.4171/PRIMS/55-1-2
State	Published - 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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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{\textquoteright}q (ŝln) 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.",

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N2 - 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 (ŝln) 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.

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On the cluster nature and quantization of geometric R-matrices

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