Yang–Baxter Equations

  • Yuri I. ManinEmail author
Part of the CRM Short Courses book series (CRMSC)


Let \(F\) be a linear space, \(R:F\otimes F\rightarrow F\otimes F\) an invertible linear map. It is well known that if \(R=S_{(12)}:f_1\otimes f_2 \mapsto f_2 \otimes f_1\), then one can define a representation of the symmetric group \({{\mathrm{S}}}_n\) on \(F^{\otimes n}\) by the following prescription: represent each element \(\sigma \in {{\mathrm{S}}}_n\) as a product of transpositions of neighbors and apply \(R_{i, i+1}=S_{(i, i+1)}\) instead of each \((i, i+1)\). Of course, such a decomposition is nonunique but the resulting linear operator does not depend on it.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Max Planck Institute for MathematicsBonnGermany

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