Generalized flag geometries associated with (2k + 1)-graded Lie algebras

Julien Chenal

paper

Generalized flag geometries associated with (2k + 1)-graded Lie algebras

arXiv:1007.4076

Abstract

In this paper, we present the construction of a geometric object, called a generalized flag geometry, $(X^{+}; X^{-})$ , corresponding to a (2k +1)-graded Lie algebra $g = g_{k} \oplus \dots \oplus g_{- k}$ . We prove that $(X^{+}; X^{-}) c anb er e a l i z e d in s i d e t h es p a ceo f inn er f i l t r a t i o n so f g an d w e u se t hi sr e a l i z a t i o n t oco n s t r u c t " a l g e b r ai c b u n d l es " o n$ X^+ $an d$ X^- $an d so m esec t i o n so f t h ese b u n d l es . T hank s t o t h eseco n s t r u c t i o n s, w ec an g i v e a r e a l i z a t i o n o f$ g $a s a L i e a l g e b r a o f p o l y n o mia l ma p so n t h e p os i t i v e p a r t o f$ g $,$ n^+_1:=g_1\oplus\dots\oplus g_k $, an d a t r i v ia l i z a t i o nin$ n^+_1 $o f t h e a c t i o n o f t h e g r o u p o f a u t o m or p hi s m so f$ g$ by "birational"maps.