On the cluster nature and quantization of geometric $R$-matrices

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 $U_q^{\prime }({\widehat{\mathfrak{sl}}}_n)$ 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.

42 pages

References in corpus (2)

• The quantum dilogarithm and representations quantum cluster varieties
• (Super)Yang-Mills at Finite Heavy-Quark Density

Cited by in corpus (2)

• Toric networks, geometric $R$-matrices and generalized discrete Toda lattices
• Symmetric Group Action of the Birational $R$-matrix