Vol.39 (3) 2017 — 3 articles found.

Tilings Defined by Root Systems of Kac-Moody Algebras

C. R. Math. Rep. Acad. Sci. Canada Vol. 39 (3) 2017, pp. 103-115
Vol.39 (3) 2017
Yuan Yao Details
(Received: 2017-06-06 , Revised: 2017-06-07 )
(Received: 2017-06-06 , Revised: 2017-06-07 )

Yuan Yao,University of Toronto, Department of Mathematics, 40 St George Street, Toronto, Ontario, Canada M4S 2E4; e-mail: yy.yao@mail.utoronto.ca


For root systems of symmetrizable Kac-Moody algebras, we study a tiling of the positive root cone of the form \(\bigcup_{w\in W} (1-w) C ^+\), where \(W\) is the Weyl group and \(C^+\) is the fundamental chamber. We show for general symmetrizable Kac-Moody algebras the tiles are disjoint, and the gaps between top dimensional tiles have codimension \(\geq2\). For affine Kac-Moody algebras we completely describe the closure \(\bigcup_{w\in W} \overline{(1-w) C ^+}\).

Pour les systèmes des racines d’algèbres de Kac-Moody symétriques, nous étudions un carrelage du cône des racines positifs de la forme \(\bigcup_{w\in W} (1-w) C ^+\), où \(W\) est le groupe de Weyl et \( C^+ \) est la chambre fondamentale. Nous montrons que les carreaux sont disjoints pour les algèbres de Kac-Moody symétriques, et les lacunes entre les carreaux de dimension supérieure ont codimension \(\geq 2\). Pour les algèbres de Kac-Moody affines, nous décrivons complètement \(\bigcup_{w\in W} \overline{(1-w) C ^+}\).

Keywords: Kac-Moody Algebras, Root System, Tiling

AMS Subject Classification: , Kac-Moody (super)algebras (structure and representation theory) 17B22, 17B67

PDF(click to download): Tilings Defined by Root Systems of Kac-Moody Algebras

Discrete Invariants of Generically Inconsistent Systems of Laurent Polynomials

C. R. Math. Rep. Acad. Sci. Canada Vol. 39 (3) 2017, pp. 90-102
Vol.39 (3) 2017
Leonid Monin Details
(Received: 2017-03-17 , Revised: 2017-04-17 )
(Received: 2017-03-17 , Revised: 2017-04-17 )

Leonid Monin,Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario, Canada M5S 2E4; e-mail: lmonin@math.toronto.edu


Let \( \mathcal{A}_1, \ldots, \mathcal{A}_k \) be finite sets in \( \mathbb{Z}^n \) and let \( Y \subset (\mathbb{C}^*)^n \) be an algebraic variety defined by a system of equations \[f_1 = \ldots = f_k = 0,\] where \( f_1, \ldots, f_k \) are Laurent polynomials with supports in \(\mathcal{A}_1, \ldots, \mathcal{A}_k\). Assuming that \( f_1, \ldots, f_k \) are sufficiently generic, the Newton polyhedron theory computes discrete invariants of \( Y \) in terms of the Newton polyhedra of \( f_1, \ldots, f_k \). It may appear that the generic system with fixed supports \( \mathcal{A}_1, \ldots, \mathcal{A}_k \) is inconsistent. In this paper, we compute discrete invariants of algebraic varieties defined by systems of equations which are generic in the set of consistent system with support in \( \mathcal{A}_1, \ldots, \mathcal{A}_k\) by reducing the question to the Newton polyhedra theory. Unlike the classical situation, not only the Newton polyhedra of \(f_1,\dots,f_k\), but also the supports \(\mathcal{A}_1,\dots,\mathcal{A}_k\) themselves appear in the answers.

Soit \( \mathcal{A}_1, \ldots, \mathcal{A}_k \) un ensemble fini dans \( \mathbb{Z}^n \) et soit \( Y \subset (\mathbb{C}^*)^n \) une variété algébrique définie par un système d’équations \[f_1 = \ldots = f_k = 0,\]\( f_1, \ldots, f_k \) sont les polynômes de Laurent avec support dans \(\mathcal{A}_1, \ldots, \mathcal{A}_k\). Supposant que \( f_1, \ldots, f_k \) soient suffisamment génériques, la théorie du polyèdre de Newton calcule les invariants discrets de \( Y \) en fonction du polyèdre de Newton de \( f_1, \ldots, f_k \). Il peut sembler que le système avec support fixe \( \mathcal{A}_1, \ldots, \mathcal{A}_k \) est inconsistent. Dans ce papier, nous calculons les invariants discrets des variétés algébriques définies par des systèmes d’équations qui sont génériques dans l’ensemble des systèmes cohérents avec support dans \( \mathcal{A}_1, \ldots, \mathcal{A}_k\) en réduisant la question à la théorie du polyèdre de Newton. Contrairement à la situation classique, non seulement le polyèdre de Newton de \(f_1,\dots,f_k\), mais aussi les supports \(\mathcal{A}_1,\dots,\mathcal{A}_k\) eux-mêmes apparaissent dans la solution.

Keywords: Laurent polynomials, Newton polyhedra, generically inconsistent systems, resultants

AMS Subject Classification: Toric varieties; Newton polyhedra 14M25

PDF(click to download): Discrete Invariants of Generically Inconsistent Systems of Laurent Polynomials

Renormalization of Unicritical Analytic Circle Maps

C. R. Math. Rep. Acad. Sci. Canada Vol. 39 (3) 2017, pp. 77-89
Vol.39 (3) 2017
Michael Yampolsky Details
(Received: 2016-09-26 , Revised: 2016-12-23 )
(Received: 2016-09-26 , Revised: 2016-12-23 )

Michael Yampolsky,Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4; e-mail: yampol@math.toronto.edu


In this paper we generalize renormalization theory for analytic critical circle maps with a cubic critical point to the case of maps with an arbitrary odd critical exponent by proving a quasiconformal rigidity statement for renormalizations of such maps.

Dans cet article on généralise la théorie de la renormalisation pour les transformations criticales analytiques du circle à point critical cubique au cas de transformations à exposant critical impair arbitraire, en démontrant une affirmation de rigidité quasi-conforme.

Keywords: Blaschke fractions, Renormalization, critical circle maps, rigidity

AMS Subject Classification: Maps of the circle, Universality; renormalization, Renormalization 37E10, 37E20, 37F25

PDF(click to download): Renormalization of Unicritical Analytic Circle Maps

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