[For the entire collection see Zbl 0646.00016.]
Motivated by the study of rational solutions of the classical Yang-Baxter equations due to A. A. Belavin and V. G. Drinfel’d [Funkts. Anal. Prilozh. 16, No.3, 1-29 (1982; Zbl 0504.22016)], a flat connection is constructed over \(X_ n=\{(z_ 1,...,z_ n)\in {\mathbb{C}}^ n\); \(z_ i\neq z_ j\) if \(i\neq j\}\), the fundamental group of which is the pure braid group on n strings. For a simple complex Lie algebra \({\mathfrak g}\) with Casimir element c and a set of n irreducible representations \(\rho_ i: {\mathfrak g}\to End(V_ i)\), one defines \(\Omega_{ij}\in End(V_ 1\otimes... \otimes V_ n)\) by \(\Omega_{ij}=(\rho_ i\otimes \rho_ j)(\Omega)\) where \(\Omega =(1/2)(\Delta c-c\otimes 1-1\otimes c)\in U({\mathfrak g})\otimes U({\mathfrak g}).\) Given a complex number \(\lambda\), a connection is then defined by \(\sum \lambda \Omega_{ij} d \log (z_ i-z_ j),\) the integrability of which follows from the infinitesimal pure braid relations. Thus the monodromy of the connection gives rise to a linear representation of the fundamental group of \(X_ n\), which becomes a linear representation of the braid group depending upon a parameter \(\lambda\) if all the representations \(\rho_ i\) are the same.
In a preceding paper of the author [Linear representations of braid groups and classical Yang-Baxter equations, Contemp. Math. 78, 339-363 (1988; Zbl 0661.20026)] the case of \({\mathfrak g}={\mathfrak sl}(2,{\mathbb{C}})\) and \(\rho_ i=the\) 2-dimensional irreducible representtion has been studied, giving rise to the Pimsner-\(Popa\)-\(Temperley\)-\(Lie\) representation. Here, the case \({\mathfrak g}={\mathfrak sl}(m+1,{\mathbb{C}})\) and \(\rho_ i=the\) \((m+1)\)- dimensional natural representation is considered. Hecke algebra representations of the braid group corresponding to Young diagrams of depth \(\leq m+1\) are obtained.