Linear representations of braid groups and classical Yang-Baxter equations. (English) Zbl 0661.20026

Kohno, Toshitake

Braids, AMS-IMS-SIAM Jt. Summer Res. Conf., Santa Cruz/Calif. 1986, Contemp. Math. 78, 339-363 (1988).

[For the entire collection see Zbl 0651.00010.]
The author studies linear representations of the pure braid groups and the braid groups. For this purpose he considers the following notations: \(B_ n:\) braid group on n strings with generators \(\sigma_ i\), \(1\leq i\leq n-1\), represented by a braid interchanging strings i and \(i+1\). \(P_ n:\) pure braid group on n strings with generators \[ \gamma_{ij}=\sigma_ i\sigma_{i+1}...\sigma_{j-1}\sigma^ 2_ j\sigma^{-1}_{j-1}...\sigma_ i^{-1},\quad 1\leq i<j\leq n. \] \(X_ n=\{(z_ 1,...,z_ n)\in {\mathbb{C}}^ n\); \(z_ i\neq z_ j\) if \(i\neq j\}\). \(\Omega =\sum_{1\leq i<j\leq n}\Omega_{ij}d\log (z_ i-z_ j)\) defined over \(X_ n\), where \(\Omega_{ij}\) are \(m\times m\) matrices with \({\mathbb{C}}\) coefficients.
After giving some characterizations of the braid groups in details, he proves the following theorems: Theorem 1: Let \(\theta\) : \(P_ n\to GL(m,{\mathbb{C}})\) be a linear representation such that each \(\| \theta (\gamma_{ij})-1\|\) is sufficiently small for \(1\leq i<j\leq n\). Then there exist constant matrices \(\Omega_{ij}\), \(1\leq i<j\leq n\), close to 0, satisfying the infinitesimal pure braid relations, such that the monodromy of the connection \(\Omega =\sum_{1\leq i<j\leq n}\Omega_{ij}d\log (z_ i-z_ j)\) is the given \(\theta\). Theorem 2: Let \(\theta_{\lambda}: B_ n\to End(V^{\otimes n})\) be the one parameter family of the monodromy representation of the connection \(\sum_{1\leq i<j\leq n}\lambda \tau_{ij}d\log (z_ 1-z_ j)\) defined above by means of \({\mathfrak g}=sl(2,{\mathbb{C}})\) and its two dimensional irreducible representation. Then \(\theta_{\lambda}\) is equivalent to the linear representation of \(B_ n\) defined by \(\theta_{\lambda}(\sigma_ i)=-q^{1/4}\{qe_ i-(1-e_ i)\}\), \(1\leq i\leq n-1\), where \(q=e^{-2\pi i\lambda}.\)
At the end of this paper the author gives some open questions concerning the subject studied. We think that this work makes some important contributions to this field. Still one can hope for some generalizations.

Reviewer: T.Başkan

MSC:

20F36	Braid groups; Artin groups
35F05	Linear first-order PDEs
55P62	Rational homotopy theory
20F38	Other groups related to topology or analysis
57M25	Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

simplicial complex; Lie algebra; affine variety; linear representations; pure braid groups; monodromy

Citations:

Zbl 0651.00010