The trigonometric $$E_8$$ E 8 R -matrix
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The trigonometric E8 R-matrix Paul Zinn-Justin1 Received: 12 May 2020 / Revised: 12 May 2020 / Accepted: 28 August 2020 © Springer Nature B.V. 2020
Abstract e8 ) in its 249-dimensional An expression for the R-matrix associated with Uq ( representation is given using the diagrammatic calculus of Uq (e8 ) invariants. Keywords Quantized affine algebras · R-matrix · Yang-Baxter equation · E8 Mathematics Subject Classification 16T25 · 82B23 · 81R12
1 Introduction Quantized affine algebras are pseudotriangular Hopf algebras [6]; as such, given any pair of “generic” representations V1 and V2 , there exists an intertwiner between V1 ⊗V2 and V2 ⊗ V1 , the so-called R-matrix. R-matrices are the central object in quantum integrable systems, and for most algebras and many low-dimensional representations, they are known explicitly [1,5,7,9,10,13,14,17,18,20]. As far as the author knows, the only case that has not been studied in detail is that of (the quantized affine algebra of) e8 . The goal of this short paper is therefore rather modest: it is the explicit computation of e8 ) in its lowest nontrivial the R-matrix associated with the quantized loop algebra Uq ( representation, which is of dimension 249 (and is not irreducible for the nonaffine algebra Uq (e8 ), which is a source of complication). In order to do so, we develop a natural diagrammatic language for the theory of invariants of Uq (e8 ) based on tensors of the adjoint representation, which we then apply to the computation of the R-matrix. The latter is fixed (up to normalization) by solving the equations that require it to be an intertwiner [8]. The main challenge of this paper is computational: for instance, the R-matrix is a matrix of size 2492 = 62001, so altogether an array of approximately 4 billion
PZJ was supported by ARC Grants FT150100232 and DP180100860. He would like to thank A. Kuniba for useful discussions and J. Lamers for comments on the manuscript.
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Paul Zinn-Justin [email protected] School of Mathematics and Statistics, The University of Melbourne, Melbourne, VIC 3010, Australia
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entries (albeit a sparse array, since only 0.05% of these are nonzero). This means that many computations in this paper cannot be performed by hand, and the help of a symbol computation program is necessary. (The author’s code is available on request.) In particular, quite a few results are presented below without proof; if so, the reader should assume that they are the result of a computer-assisted calculation. The immediate motivation for this paper came from the work of the author in collabe8 ) appeared unexpectedly oration with A. Knutson [15,16] where the R-matrix of Uq ( in the computation of structure constants of the K -theory of 4-step flag varieties. Another possible source of interest is the fact that the scaling limit of the Ising model at the critical temperature in a magnetic field is known to be related to an e8 integrable field theory [2,19], and a vertex lattice model based on e8 might be desirable. The paper is organize
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