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

TL;DR

This paper introduces a new geometric framework called generalized flag geometry for (2k+1)-graded Lie algebras, providing algebraic and geometric tools to realize the algebra as polynomial maps and study automorphisms.

Contribution

It constructs generalized flag geometries associated with (2k+1)-graded Lie algebras and develops their realization within inner filtrations, enabling polynomial map representations and automorphism analysis.

Findings

Realization of (2k+1)-graded Lie algebras as polynomial maps on positive parts.

Construction of algebraic bundles and sections on generalized flag geometries.

Trivialization of automorphism group actions via birational maps.

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^-) can be realized inside the space of inner filtrations of g and we use this realization to construct "algebraic bundles" on $X^+$ and $X^-$ and some sections of these bundles. Thanks to these constructions, we can give a realization of $g$ as a Lie algebra of polynomial maps on the positive part of $g$, $n^+_1:=g_1\oplus\dots\oplus g_k$, and a trivialization in $n^+_1$ of the action of the group of automorphisms of $g$ by "birational"maps.