A uniform trigonometric R-matrix for the exceptional series

Abstract

The exceptional series is a finite list of points on a projective line with a simple Lie algebra attached to each point. This list of Lie algebras includes the five exceptional Lie algebras. We give a uniform trigonometric $R$-matrix for the exceptional series in the representation $L\oplus I$, where $L$ is the quantum deformation of the adjoint representation and $I$ is the trivial representation. We construct a sixteen dimensional algebra, $A^\square(\mathit2)$, which interpolates the algebras $\mathrm{End}(\otimes^2(L\oplus I))$ and a 287 dimensional algebra, $A^\square(\mathit3)$, which interpolates the algebras $\mathrm{End}(\otimes^3(L\oplus I))$. The $R$-matrix lives in $A^\square(\mathit2)$ and satisfies the Yang-Baxter equation in $A^\square(\mathit3)$; it interpolates the trigonometric $R$-matrices for the points in the exceptional series.