Quantization of Bending Deformations of Polygons In , Hypergeometric Integrals and the Gassner Representation

Kapovich, Michael ; Millson, John J.

Canadian mathematical bulletin, Tome 44 (2001) no. 1, pp. 36-60

Résumé

The Hamiltonian potentials of the bending deformations of $n$ -gons in ${{\mathbb{E}}^{3}}$ studied in $\left[ \text{KM} \right]$ and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra ${{P}_{n}}$ of the pure braid group ${{P}_{n}}$ on the moduli space ${{M}_{r}}$ of $n$ -gon linkages with the side-lengths $r\,=\,\left( {{r}_{1}},\ldots ,{{r}_{n}} \right)$ in ${{\mathbb{E}}^{3}}$ . If $e\,\in \,{{M}_{r}}$ is a singular point we may linearize the vector fields in ${{P}_{n}}$ at $e$ . This linearization yields a flat connection $\nabla$ on the space $\mathbb{C}_{*}^{n}$ of $n$ distinct points on $\mathbb{C}$ . We show that the monodromy of $\nabla$ is the dual of a quotient of a specialized reduced Gassner representation.

Détail

DOI : 10.4153/CMB-2001-006-3

Mots-clés : 53D30, 53D50

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