TY - JOUR
T1 - Reflectable bases for affine reflection systems
AU - Azam, Saeid
AU - Yamane, Hiroyuki
AU - Yousofzadeh, Malihe
N1 - Funding Information:
E-mail addresses: [email protected] (S. Azam), [email protected] (H. Yamane), [email protected] (M. Yousofzadeh). 1 This research was in part supported by a grant from IPM (Nos. 89170216, 89170030). The authors would like to thank the Center of Excellence for Mathematics, University of Isfahan.
PY - 2012
Y1 - 2012
N2 - The notion of a "root base" together with its geometry plays a crucial role in the theory of finite and affine Lie theory. However, it is known that such a notion does not exist for the recent generalizations of finite and affine root systems such as extended affine root systems and affine reflection systems. In this work, we consider the notion of a "reflectable base" for an affine reflection system R. A reflectable base for R is a minimal subset Π of roots such that the non-isotropic part of the root system can be recovered by reflecting roots of Π relative to the hyperplanes determined by Π. We give a full characterization of reflectable bases for tame irreducible affine reflection systems of reduced types, excluding types E 6,7,8. As a by-product of our results, we show that if the root system under consideration is locally finite, then any reflectable base is an integral base.
AB - The notion of a "root base" together with its geometry plays a crucial role in the theory of finite and affine Lie theory. However, it is known that such a notion does not exist for the recent generalizations of finite and affine root systems such as extended affine root systems and affine reflection systems. In this work, we consider the notion of a "reflectable base" for an affine reflection system R. A reflectable base for R is a minimal subset Π of roots such that the non-isotropic part of the root system can be recovered by reflecting roots of Π relative to the hyperplanes determined by Π. We give a full characterization of reflectable bases for tame irreducible affine reflection systems of reduced types, excluding types E 6,7,8. As a by-product of our results, we show that if the root system under consideration is locally finite, then any reflectable base is an integral base.
KW - Affine reflection systems
KW - Extended affine Weyl groups
KW - Extended affine root systems
KW - Reflectable bases
UR - http://www.scopus.com/inward/record.url?scp=84865319670&partnerID=8YFLogxK
U2 - 10.1016/j.jalgebra.2012.08.004
DO - 10.1016/j.jalgebra.2012.08.004
M3 - 学術論文
AN - SCOPUS:84865319670
SN - 0021-8693
VL - 371
SP - 63
EP - 93
JO - Journal of Algebra
JF - Journal of Algebra
ER -