The geometry of dented pentagram maps

Khesin, Boris, and Soloviev, Fedor. "The geometry of dented pentagram maps." Journal of the European Mathematical Society 018.1 (2016): 147-179. <http://eudml.org/doc/277382>.

@article{Khesin2016,
abstract = {We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension $d$ there are $d–1$ such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps with those for the higher pentagram maps constructed with the help of short diagonal hyperplanes. When restricted to corrugated polygons, the dented pentagram maps coincide with one another and with the corresponding corrugated pentagram map. Finally, we prove integrability for a variety of pentagram maps for generic and partially corrugated polygons in higher dimensions.},
author = {Khesin, Boris, Soloviev, Fedor},
journal = {Journal of the European Mathematical Society},
keywords = {pentagram maps; space polygons; Lax representation; discrete integrable system; KdV hierarchy; Boussinesq equation; algebraic-geometric integrability; pentagram maps; space polygons; Lax representation; discrete integrable system; KdV hierarchy; Boussinesq equation; algebraic-geometric integrability},
language = {eng},
number = {1},
pages = {147-179},
publisher = {European Mathematical Society Publishing House},
title = {The geometry of dented pentagram maps},
url = {http://eudml.org/doc/277382},
volume = {018},
year = {2016},
}

TY - JOUR
AU - Khesin, Boris
AU - Soloviev, Fedor
TI - The geometry of dented pentagram maps
JO - Journal of the European Mathematical Society
PY - 2016
PB - European Mathematical Society Publishing House
VL - 018
IS - 1
SP - 147
EP - 179
AB - We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension $d$ there are $d–1$ such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps with those for the higher pentagram maps constructed with the help of short diagonal hyperplanes. When restricted to corrugated polygons, the dented pentagram maps coincide with one another and with the corresponding corrugated pentagram map. Finally, we prove integrability for a variety of pentagram maps for generic and partially corrugated polygons in higher dimensions.
LA - eng
KW - pentagram maps; space polygons; Lax representation; discrete integrable system; KdV hierarchy; Boussinesq equation; algebraic-geometric integrability; pentagram maps; space polygons; Lax representation; discrete integrable system; KdV hierarchy; Boussinesq equation; algebraic-geometric integrability
UR - http://eudml.org/doc/277382
ER -