Integrable connections related to Manin and Schechtman's higher braid groups (Q1813746)

Denote by \(D_ J\) the hyperplane of \(\mathbb{C}^ n\) labelled with \(J\subset\{1,2,\dots,n\}\) such that \(| J| =k\), \(k\leq n\). The fundamental group of \(U(n,k)=\mathbb{C}^ n-\bigcup_ J D_ J\), called higher braid group, has been studied by \textit{Yu. I. Manin} and \textit{V. V. Shekhtman} [Algebraic number theory --- in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values \(L\)-Funct., Berkeley/CA 1987, Adv. Stud. Pure Math. 17, 289-308 (1989)]. Using also \textit{K. T. Chen's} results [Adv. Math. 23, 181-210 (1977; Zbl 0345.58003); Bull. Am. Math. Soc. 83, 831-879 (1977; Zbl 0389.58001)], it follows that the completion of the group ring of \(U(n,k)\) with respect to the powers of the augmentation ideal is generated by \(X_ J\), which are in one-to-one correspondence with the planes \(D_ J\), with the relations \[ [X_ J,\sum_{I\subset K}X_ I]=0 \quad \text{ for any }K\subset\{1,2,\dots,n\}\text{ with }| K|=k+2;\leqno(i) \] \[ [X_{J_ 1},X_{J_ 2}]=0\quad \text{if }| J_ 1\cup J_ 2|\geq k+3.\leqno(ii) \] They give the integrability condition for the connection of the form \(\sum_ J X_ J d\log\varphi_ J\), where \(X_ J\) is a constant matrix and \(\varphi_ J\) is a linear form with \(\text{Ker }\varphi_ J=D_ J\). For \(k=1\) these relations were studied in relation with the classical Yang-Baxter equation and a one-parameter family of linear representations of the pure braid groups as the holonomy of this connection for any simple Lie algebra and its representations was obtained [\textit{T. Kohno}, Ann. Inst. Fourier 37, No. 4, 139-160 (1987; Zbl 0634.58040)]. The author gives a generalization of this construction to \(U(n,k)\) with \(k\geq 2\). In fact, given a finite-dimensional complex simple Lie algebra and its irreducible representations, he constructs a one-parameter family of integrable connections over \(U(n,k)\). The holonomy of these connections give linear representations of the higher braid groups.