On the Cluster Nature and Quantization of Geometric $R$-Matrices

Rei Inoue; Thomas Lam; Pavlo Pylyavskyy

doi:10.4171/prims/55-1-2

JournalsprimsVol. 55, No. 1pp. 25–78

On the Cluster Nature and Quantization of Geometric $R$ -Matrices

Rei Inoue
Chiba University, Japan
Thomas Lam
University of Michigan, Ann Arbor, USA
Pavlo Pylyavskyy
University of Minnesota, Minneapolis, USA

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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 $U_{q}^{'} (\hat{sl}_{n})$ 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.

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Rei Inoue, Thomas Lam, Pavlo Pylyavskyy, On the Cluster Nature and Quantization of Geometric $R$ -Matrices. Publ. Res. Inst. Math. Sci. 55 (2019), no. 1, pp. 25–78

DOI 10.4171/PRIMS/55-1-2