Intertwiners of representations of untwisted quantum affine algebras and Yangians

What is it about?

We provide a new method of computing the R-matrices using the q-characters. These matrices arise from intertwiners or module homomorphisms acting in tensor squares of representations of quantum affine algebras (in the trigonometric case) and Yangians (in the rational case). The method is then applied to compute the R-matrices corresponding to the first fundamental representation of all types of untwisted quantum affine algebras. The Yangian R-matrices are obtained as a limit of the ones for quantum affine algebras. The most challenging case is that of E8 where the tensor square has non-trivial multiplicities, and the size of the representation is 249-dimensional making the R-matrix a 62,001 by 62,001 matrix having close to 2 billion entries.

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Photo by Emmie Van on Unsplash

Photo by Emmie Van on Unsplash

Why is it important?

The R-matrices are solutions of the quantum Yang-Baxter equation, which plays a central role in statistical mechanics, knot theory, integrable systems, stochastic vertex models and mathematical physics in general. They provide representations of the braid equation, depending on the spectral parameter z and the quantization parameter q. The Yang-Baxter equation is a sign of integrability and almost co-commutativity, simplifying analysis of physical models possessing such a property.